How can we design proteins with desired functions? We are motivated by a chemical intuition that both geometric structure and biochemical properties are critical to a protein's function. In this paper, we propose SurfPro, a new method to generate functional proteins given a desired surface and its associated biochemical properties. SurfPro comprises a hierarchical encoder that progressively models the geometric shape and biochemical features of a protein surface, and an autoregressive decoder to produce an amino acid sequence. We evaluate SurfPro on a standard inverse folding benchmark CATH 4.2 and two functional protein design tasks: protein binder design and enzyme design. Our SurfPro consistently surpasses previous state-of-the-art inverse folding methods, achieving a recovery rate of 57.78% on CATH 4.2 and higher success rates in terms of protein-protein binding and enzyme-substrate interaction scores