In recent years it has become clear that electronic Berry curvature (BC) is a key concept to understand and predict physical properties of crystalline materials. A wealth of interesting Hall-type responses in charge, spin and heat transport are caused by the BC associated to electronic bands inside a solid: anomalous Hall effects in magnetic materials, and various nonlinear Hall and Nernst effects in non-magnetic systems that lack inversion symmetry. However, for the largest class of known materials –non-magnetic ones with inversion symmetry– electronic BC is strictly zero. Here we show that precisely for these bulk BC-free materials, a finite BC can emerge at their surfaces and interfaces. This immediately activates certain surfaces in producing Hall-type transport responses. We demonstrate this by first principles calculations of the BC at bismuth, mercury-telluride (HgTe) and rhodium surfaces of various symmetries, revealing the presence of a surface Berry curvature dipole and associated quantum nonlinear Hall effects at a number of these. This opens up a plethora of materials to explore and harness the physical effects emerging from the electronic Berry curvature associated exclusively to their boundaries.