diff --git a/include/boost/histogram/algorithm/project.hpp b/include/boost/histogram/algorithm/project.hpp
index 90ad8cd9..bff48f9d 100644
--- a/include/boost/histogram/algorithm/project.hpp
+++ b/include/boost/histogram/algorithm/project.hpp
@@ -35,7 +35,8 @@ namespace algorithm {
   histogram is summed over the removed axes.
 */
 template <class A, class S, unsigned N, typename... Ns>
-auto project(const histogram<A, S>& h, std::integral_constant<unsigned, N>, Ns...) {
+auto project(const histogram<A, S>& h, coverage cov, std::integral_constant<unsigned, N>,
+             Ns...) {
   using LN = mp11::mp_list<std::integral_constant<unsigned, N>, Ns...>;
   static_assert(mp11::mp_is_set<LN>::value, "indices must be unique");
 
@@ -53,7 +54,7 @@ auto project(const histogram<A, S>& h, std::integral_constant<unsigned, N>, Ns..
   using A2 = decltype(axes);
   auto result = histogram<A2, S>(std::move(axes), detail::make_default(old_storage));
   auto idx = detail::make_stack_buffer<int>(unsafe_access::axes(result));
-  for (auto&& x : indexed(h, coverage::all)) {
+  for (auto&& x : indexed(h, cov)) {
     auto i = idx.begin();
     mp11::mp_for_each<LN>([&i, &x](auto J) { *i++ = x.index(J); });
     result.at(idx) += *x;
@@ -61,6 +62,16 @@ auto project(const histogram<A, S>& h, std::integral_constant<unsigned, N>, Ns..
   return result;
 }
 
+/**
+  Returns a lower-dimensional histogram, summing over removed axes.
+
+  This overload uses coverage::all by default.
+*/
+template <class A, class S, unsigned N, typename... Ns>
+auto project(const histogram<A, S>& h, std::integral_constant<unsigned, N> n, Ns... ns) {
+  return project(h, coverage::all, n, ns...);
+}
+
 /**
   Returns a lower-dimensional histogram, summing over removed axes.
 
@@ -68,7 +79,7 @@ auto project(const histogram<A, S>& h, std::integral_constant<unsigned, N>, Ns..
   indices.
 */
 template <class A, class S, class Iterable, class = detail::requires_iterable<Iterable>>
-auto project(const histogram<A, S>& h, const Iterable& c) {
+auto project(const histogram<A, S>& h, coverage cov, const Iterable& c) {
   using namespace boost::mp11;
   const auto& old_axes = unsafe_access::axes(h);
 
@@ -88,7 +99,7 @@ auto project(const histogram<A, S>& h, const Iterable& c) {
   auto result =
       histogram<decltype(axes), S>(std::move(axes), detail::make_default(old_storage));
   auto idx = detail::make_stack_buffer<int>(unsafe_access::axes(result));
-  for (auto&& x : indexed(h, coverage::all)) {
+  for (auto&& x : indexed(h, cov)) {
     auto i = idx.begin();
     for (auto d : c) *i++ = x.index(d);
     result.at(idx) += *x;
@@ -97,6 +108,16 @@ auto project(const histogram<A, S>& h, const Iterable& c) {
   return result;
 }
 
+/**
+  Returns a lower-dimensional histogram, summing over removed axes.
+
+  This overload uses coverage::all by default.
+*/
+template <class A, class S, class Iterable, class = detail::requires_iterable<Iterable>>
+auto project(const histogram<A, S>& h, const Iterable& c) {
+  return project(h, coverage::all, c);
+}
+
 } // namespace algorithm
 } // namespace histogram
 } // namespace boost
diff --git a/test/algorithm_project_test.cpp b/test/algorithm_project_test.cpp
index e9fd3753..88b997f7 100644
--- a/test/algorithm_project_test.cpp
+++ b/test/algorithm_project_test.cpp
@@ -8,6 +8,7 @@
 #include <boost/histogram/algorithm/project.hpp>
 #include <boost/histogram/algorithm/sum.hpp>
 #include <boost/histogram/axis/integer.hpp>
+#include <boost/histogram/axis/regular.hpp>
 #include <boost/histogram/axis/ostream.hpp>
 #include <boost/histogram/literals.hpp>
 #include <boost/histogram/ostream.hpp>
@@ -65,6 +66,17 @@ void run_tests() {
     BOOST_TEST_EQ(hyx.at(0, 1), 1);
     BOOST_TEST_EQ(hyx.at(1, 1), 1);
     BOOST_TEST_EQ(hyx.at(2, 1), 2);
+
+    // explicit coverage::all should match default behavior
+    auto hx_all = project(h, coverage::all, 0_c);
+    BOOST_TEST_EQ(sum(hx_all), sum(hx));
+
+    // coverage::inner excludes underflow/overflow; for integer axes with default
+    // options there are no underflow/overflow bins, so inner gives same result
+    auto hx_inner = project(h, coverage::inner, 0_c);
+    BOOST_TEST_EQ(sum(hx_inner), 6);
+    BOOST_TEST_EQ(hx_inner.at(0), 2);
+    BOOST_TEST_EQ(hx_inner.at(1), 4);
   }
 
   {
@@ -134,6 +146,35 @@ void run_tests() {
     BOOST_TEST_EQ(h_210.at(2, 0, 0), 1);
     BOOST_TEST_EQ(h_210.at(2, 0, 1), 1);
   }
+
+  {
+    // regular axes have underflow/overflow by default
+    auto h = make(Tag(), axis::regular<>(2, 0.0, 2.0), axis::regular<>(3, 0.0, 3.0));
+    h(-1, -1); // underflow,underflow
+    h(0.5, 0.5);
+    h(0.5, 1.5);
+    h(1.5, 0.5);
+    h(1.5, 1.5);
+    h(1.5, 2.5); // overflow on second axis
+    h(2.5, 0.5); // overflow on first axis
+
+    auto hx_all = project(h, coverage::all, 0_c);
+    BOOST_TEST_EQ(hx_all.rank(), 1);
+    // all bins summed: 7 entries total
+    BOOST_TEST_EQ(sum(hx_all), 7);
+    // hx has underflow + 2 bins + overflow
+    BOOST_TEST_EQ(hx_all.at(-1), 1);
+    BOOST_TEST_EQ(hx_all.at(0), 2);
+    BOOST_TEST_EQ(hx_all.at(1), 3);
+    BOOST_TEST_EQ(hx_all.at(2), 1);
+
+    auto hx_inner = project(h, coverage::inner, 0_c);
+    BOOST_TEST_EQ(hx_inner.rank(), 1);
+    // only inner bins of original histogram are summed
+    BOOST_TEST_EQ(sum(hx_inner), 5);
+    BOOST_TEST_EQ(hx_inner.at(0), 2);
+    BOOST_TEST_EQ(hx_inner.at(1), 3);
+  }
 }
 
 // split out dynamic tests as workaround for compiler bug in
@@ -179,6 +220,19 @@ void run_dynamic_tests() {
     BOOST_TEST_EQ(hyx.at(1, 1), 1);
     BOOST_TEST_EQ(hyx.at(2, 1), 2);
 
+    // explicit coverage::all should match default behavior
+    x = {0};
+    auto hx_all = project(h, coverage::all, x);
+    BOOST_TEST_EQ(sum(hx_all), sum(hx));
+
+    // coverage::inner excludes underflow/overflow; for integer axes with default
+    // options there are no underflow/overflow bins, so inner gives same result
+    x = {0};
+    auto hx_inner = project(h, coverage::inner, x);
+    BOOST_TEST_EQ(sum(hx_inner), 6);
+    BOOST_TEST_EQ(hx_inner.at(0), 2);
+    BOOST_TEST_EQ(hx_inner.at(1), 4);
+
     // indices must be unique
     x = {0, 0};
     BOOST_TEST_THROWS((void)project(h, x), std::invalid_argument);