diff --git a/include/gsl/gsl_util b/include/gsl/gsl_util
index d1f7f33..53141e6 100644
--- a/include/gsl/gsl_util
+++ b/include/gsl/gsl_util
@@ -49,7 +49,7 @@ namespace gsl
 //
 
 // index type for all container indexes/subscripts/sizes
-using index = std::ptrdiff_t;
+using index = std::size_t;
 
 // final_action allows you to ensure something gets run at the end of a scope
 template <class F>
@@ -129,8 +129,8 @@ GSL_SUPPRESS(bounds.4) // NO-FORMAT: attribute
 GSL_SUPPRESS(bounds.2) // NO-FORMAT: attribute
 constexpr T& at(T (&arr)[N], const index i)
 {
-    Expects(i >= 0 && i < narrow_cast<index>(N));
-    return arr[narrow_cast<std::size_t>(i)];
+    Expects(i >= 0 && i < N);
+    return arr[i];
 }
 
 template <class Cont>
@@ -138,16 +138,15 @@ GSL_SUPPRESS(bounds.4) // NO-FORMAT: attribute
 GSL_SUPPRESS(bounds.2) // NO-FORMAT: attribute
 constexpr auto at(Cont& cont, const index i) -> decltype(cont[cont.size()])
 {
-    Expects(i >= 0 && i < narrow_cast<index>(cont.size()));
-    using size_type = decltype(cont.size());
-    return cont[narrow_cast<size_type>(i)];
+    Expects(i >= 0 && i < cont.size());
+    return cont[i];
 }
 
 template <class T>
 GSL_SUPPRESS(bounds.1) // NO-FORMAT: attribute
 constexpr T at(const std::initializer_list<T> cont, const index i)
 {
-    Expects(i >= 0 && i < narrow_cast<index>(cont.size()));
+    Expects(i >= 0 && i < cont.size());
     return *(cont.begin() + i);
 }
 
diff --git a/include/gsl/span b/include/gsl/span
index 5049b97..548587b 100644
--- a/include/gsl/span
+++ b/include/gsl/span
@@ -552,16 +552,6 @@ public:
         return data()[size() - 1];
     }
 
-    // at and operator() are deprecated to align to the public member functions of std::span
-    [[deprecated("Use operator[]")]] constexpr reference at(size_type idx) const noexcept
-    {
-        return this->operator[](idx);
-    }
-    [[deprecated("Use operator[]")]] constexpr reference operator()(size_type idx) const noexcept
-    {
-        return this->operator[](idx);
-    }
-
     constexpr pointer data() const noexcept { return storage_.data(); }
 
     // [span.iter], span iterator support
@@ -817,67 +807,67 @@ constexpr ElementType& at(span<ElementType, Extent> s, index i)
 
 // [span.obs] Free observer functions
 template <class ElementType, std::size_t Extent>
-constexpr typename span<ElementType, Extent>::size_type
-ssize(const span<ElementType, Extent>& span) noexcept
+constexpr std::ptrdiff_t
+ssize(const span<ElementType, Extent>& s) noexcept
 {
-    return span.size();
+    return static_cast<std::ptrdiff_t>(s.size());
 }
 
 // [span.iter] Free functions for begin/end functions
 template <class ElementType, std::size_t Extent>
 constexpr typename span<ElementType, Extent>::iterator
-begin(const span<ElementType, Extent>& span) noexcept
+begin(const span<ElementType, Extent>& s) noexcept
 {
-    return span.begin();
+    return s.begin();
 }
 
 template <class ElementType, std::size_t Extent = dynamic_extent>
 constexpr typename span<ElementType, Extent>::iterator
-end(const span<ElementType, Extent>& span) noexcept
+end(const span<ElementType, Extent>& s) noexcept
 {
-    return span.end();
+    return s.end();
 }
 
 template <class ElementType, std::size_t Extent>
 constexpr typename span<ElementType, Extent>::const_iterator
-cbegin(const span<ElementType, Extent>& span) noexcept
+cbegin(const span<ElementType, Extent>& s) noexcept
 {
-    return span.cbegin();
+    return s.cbegin();
 }
 
 template <class ElementType, std::size_t Extent>
 constexpr typename span<ElementType, Extent>::const_iterator
-cend(const span<ElementType, Extent>& span) noexcept
+cend(const span<ElementType, Extent>& s) noexcept
 {
-    return span.cend();
+    return s.cend();
 }
 
 template <class ElementType, std::size_t Extent>
 constexpr typename span<ElementType, Extent>::reverse_iterator
-rbegin(const span<ElementType, Extent>& span) noexcept
+rbegin(const span<ElementType, Extent>& s) noexcept
 {
-    return span.rbegin();
+    return s.rbegin();
 }
 
 template <class ElementType, std::size_t Extent>
 constexpr typename span<ElementType, Extent>::reverse_iterator
-rend(const span<ElementType, Extent>& span) noexcept
+rend(const span<ElementType, Extent>& s) noexcept
 {
-    return span.rend();
+    return s.rend();
 }
 
 template <class ElementType, std::size_t Extent>
 constexpr typename span<ElementType, Extent>::const_reverse_iterator
-crbegin(const span<ElementType, Extent>& span) noexcept
+crbegin(const span<ElementType, Extent>& s) noexcept
 {
-    return span.crbegin();
+    return s.crbegin();
 }
 
 template <class ElementType, std::size_t Extent>
 constexpr typename span<ElementType, Extent>::const_reverse_iterator
-crend(const span<ElementType, Extent>& span) noexcept
+crend(const span<ElementType, Extent>& s) noexcept
 {
-    return span.crend();
+    return s.crend();
 }
 
 } // namespace gsl
diff --git a/tests/at_tests.cpp b/tests/at_tests.cpp
index be2c7b8..0580301 100644
--- a/tests/at_tests.cpp
+++ b/tests/at_tests.cpp
@@ -33,7 +33,7 @@ TEST(at_tests, static_array)
     int a[4] = {1, 2, 3, 4};
     const int(&c_a)[4] = a;
 
-    for (int i = 0; i < 4; ++i) {
+    for (std::size_t i = 0; i < 4; ++i) {
         EXPECT_TRUE(&gsl::at(a, i) == &a[i]);
         EXPECT_TRUE(&gsl::at(c_a, i) == &a[i]);
     }
@@ -43,9 +43,7 @@ TEST(at_tests, static_array)
         std::abort();
     });
 
-    EXPECT_DEATH(gsl::at(a, -1), deathstring);
     EXPECT_DEATH(gsl::at(a, 4), deathstring);
-    EXPECT_DEATH(gsl::at(c_a, -1), deathstring);
     EXPECT_DEATH(gsl::at(c_a, 4), deathstring);
 }
 
@@ -54,7 +52,7 @@ TEST(at_tests, std_array)
     std::array<int, 4> a = {1, 2, 3, 4};
     const std::array<int, 4>& c_a = a;
 
-    for (int i = 0; i < 4; ++i) {
+    for (std::size_t i = 0; i < 4; ++i) {
         EXPECT_TRUE(&gsl::at(a, i) == &a[static_cast<std::size_t>(i)]);
         EXPECT_TRUE(&gsl::at(c_a, i) == &a[static_cast<std::size_t>(i)]);
     }
@@ -64,9 +62,7 @@ TEST(at_tests, std_array)
         std::abort();
     });
 
-    EXPECT_DEATH(gsl::at(a, -1), deathstring);
     EXPECT_DEATH(gsl::at(a, 4), deathstring);
-    EXPECT_DEATH(gsl::at(c_a, -1), deathstring);
     EXPECT_DEATH(gsl::at(c_a, 4), deathstring);
 }
 
@@ -75,7 +71,7 @@ TEST(at_tests, std_vector)
     std::vector<int> a = {1, 2, 3, 4};
     const std::vector<int>& c_a = a;
 
-    for (int i = 0; i < 4; ++i) {
+    for (std::size_t i = 0; i < 4; ++i) {
         EXPECT_TRUE(&gsl::at(a, i) == &a[static_cast<std::size_t>(i)]);
         EXPECT_TRUE(&gsl::at(c_a, i) == &a[static_cast<std::size_t>(i)]);
     }
@@ -85,9 +81,7 @@ TEST(at_tests, std_vector)
         std::abort();
     });
 
-    EXPECT_DEATH(gsl::at(a, -1), deathstring);
     EXPECT_DEATH(gsl::at(a, 4), deathstring);
-    EXPECT_DEATH(gsl::at(c_a, -1), deathstring);
     EXPECT_DEATH(gsl::at(c_a, 4), deathstring);
 }
 
@@ -96,8 +90,8 @@ TEST(at_tests, InitializerList)
     const std::initializer_list<int> a = {1, 2, 3, 4};
 
     for (int i = 0; i < 4; ++i) {
-        EXPECT_TRUE(gsl::at(a, i) == i + 1);
-        EXPECT_TRUE(gsl::at({1, 2, 3, 4}, i) == i + 1);
+        EXPECT_TRUE(gsl::at(a, static_cast<std::size_t>(i)) == i + 1);
+        EXPECT_TRUE(gsl::at({1, 2, 3, 4}, static_cast<std::size_t>(i)) == i + 1);
     }
 
     std::set_terminate([] {
@@ -105,9 +99,7 @@ TEST(at_tests, InitializerList)
         std::abort();
     });
 
-    EXPECT_DEATH(gsl::at(a, -1), deathstring);
     EXPECT_DEATH(gsl::at(a, 4), deathstring);
-    EXPECT_DEATH(gsl::at({1, 2, 3, 4}, -1), deathstring);
     EXPECT_DEATH(gsl::at({1, 2, 3, 4}, 4), deathstring);
 }
 
@@ -120,12 +112,12 @@ static constexpr bool test_constexpr()
     const std::array<int, 4>& c_a2 = a2;
 
     for (int i = 0; i < 4; ++i) {
-        if (&gsl::at(a1, i) != &a1[i]) return false;
-        if (&gsl::at(c_a1, i) != &a1[i]) return false;
+        if (&gsl::at(a1, static_cast<std::size_t>(i)) != &a1[i]) return false;
+        if (&gsl::at(c_a1, static_cast<std::size_t>(i)) != &a1[i]) return false;
         // requires C++17:
         // if (&gsl::at(a2, i) != &a2[static_cast<std::size_t>(i)]) return false;
-        if (&gsl::at(c_a2, i) != &c_a2[static_cast<std::size_t>(i)]) return false;
-        if (gsl::at({1, 2, 3, 4}, i) != i + 1) return false;
+        if (&gsl::at(c_a2, static_cast<std::size_t>(i)) != &c_a2[static_cast<std::size_t>(i)]) return false;
+        if (gsl::at({1, 2, 3, 4}, static_cast<std::size_t>(i)) != i + 1) return false;
     }
 
     return true;
diff --git a/tests/span_tests.cpp b/tests/span_tests.cpp
index 99ee0d8..ce10325 100644
--- a/tests/span_tests.cpp
+++ b/tests/span_tests.cpp
@@ -25,7 +25,7 @@
 #include <iterator>    // for reverse_iterator, operator-, operator==
 #include <memory>      // for unique_ptr, shared_ptr, make_unique, allo...
 #include <regex>       // for match_results, sub_match, match_results<>...
-#include <stddef.h>    // for ptrdiff_t
+#include <cstddef>     // for ptrdiff_t
 #include <string>      // for string
 #include <type_traits> // for integral_constant<>::value, is_default_co...
 #include <vector>      // for vector
@@ -161,11 +161,7 @@ TEST(span_test, from_pointer_length_constructor)
                 EXPECT_TRUE(s.data() == &arr[0]);
                 EXPECT_TRUE(s.empty() == (i == 0));
                 for (std::size_t j = 0; j < i; ++j)
-                {
                     EXPECT_TRUE(arr[j] == s[j]);
-                    EXPECT_TRUE(arr[j] == s.at(j));
-                    EXPECT_TRUE(arr[j] == s(j));
-                }
             }
             {
                 span<int> s = {&arr[i], 4 - i};
@@ -174,11 +170,7 @@ TEST(span_test, from_pointer_length_constructor)
                 EXPECT_TRUE(s.empty() == ((4 - i) == 0));
 
                 for (std::size_t j = 0; j < 4 - i; ++j)
-                {
                     EXPECT_TRUE(arr[j + i] == s[j]);
-                    EXPECT_TRUE(arr[j + i] == s.at(j));
-                    EXPECT_TRUE(arr[j + i] == s(j));
-                }
             }
         }
     }
@@ -976,52 +968,6 @@ TEST(span_test, from_array_constructor)
      }
  }
 
- TEST(span_test, at_call)
- {
-    std::set_terminate([] {
-        std::cerr << "Expected Death. at_call";
-        std::abort();
-    });
-     int arr[4] = {1, 2, 3, 4};
-
-     {
-         span<int> s = arr;
-         EXPECT_TRUE(s.at(0) == 1);
-         EXPECT_DEATH(s.at(5), deathstring);
-     }
-
-     {
-         int arr2d[2] = {1, 6};
-         span<int, 2> s = arr2d;
-         EXPECT_TRUE(s.at(0) == 1);
-         EXPECT_TRUE(s.at(1) == 6);
-         EXPECT_DEATH(s.at(2), deathstring);
-     }
- }
-
- TEST(span_test, operator_function_call)
- {
-    std::set_terminate([] {
-        std::cerr << "Expected Death. operator_function_call";
-        std::abort();
-    });
-     int arr[4] = {1, 2, 3, 4};
-
-     {
-         span<int> s = arr;
-         EXPECT_TRUE(s(0) == 1);
-         EXPECT_DEATH(s(5), deathstring);
-     }
-
-     {
-         int arr2d[2] = {1, 6};
-         span<int, 2> s = arr2d;
-         EXPECT_TRUE(s(0) == 1);
-         EXPECT_TRUE(s(1) == 6);
-         EXPECT_DEATH(s(2), deathstring);
-     }
- }
-
  TEST(span_test, iterator_default_init)
  {
      span<int>::iterator it1;
@@ -1093,8 +1039,8 @@ TEST(span_test, from_array_constructor)
      int a[] = {1, 2, 3, 4};
      span<int> s{a};
 
-     EXPECT_TRUE((std::is_same<decltype(s.size()), decltype(ssize(s))>::value));
-     EXPECT_TRUE(s.size() == ssize(s));
+     EXPECT_FALSE((std::is_same<decltype(s.size()), decltype(ssize(s))>::value));
+     EXPECT_TRUE(s.size() == static_cast<std::size_t>(ssize(s)));
  }
 
  TEST(span_test, iterator_comparisons)
diff --git a/tests/utils_tests.cpp b/tests/utils_tests.cpp
index b6b5fc9..78bf59d 100644
--- a/tests/utils_tests.cpp
+++ b/tests/utils_tests.cpp
@@ -23,6 +23,7 @@
 #include <limits>      // for numeric_limits
 #include <stdint.h>    // for uint32_t, int32_t
 #include <type_traits> // for is_same
+#include <cstddef>     // for std::size_t
 
 using namespace gsl;
 
@@ -37,7 +38,7 @@ void g() { j += 1; }
 
 TEST(utils_tests, sanity_check_for_gsl_index_typedef)
 {
-    static_assert(std::is_same<gsl::index, std::ptrdiff_t>::value,
+    static_assert(std::is_same<gsl::index, std::size_t>::value,
                   "gsl::index represents wrong arithmetic type");
 }