The COVID-19 pandemic has triggered concerns about the emergence of more infectious and pathogenic viral strains. As a public health measure, efficient screening methods are needed to determine the functional effects of new sequence variants. Here we show that structural modeling of SARS-CoV-2 Spike protein binding to the human ACE2 receptor, the first step in host-cell entry, predicts many novel variant combinations with enhanced binding affinities. By focusing on natural variants at the Spike-hACE2 interface and assessing over 700 mutant complexes, our analysis reveals that high-affinity Spike mutations (including N440K, S443A, G476S, E484R, G502P) tend to cluster near known human ACE2 recognition sites (K31 and K353). These Spike regions are structurally flexible, allowing certain mutations to optimize interface interaction energies. Although most human ACE2 variants tend to weaken binding affinity, they can interact with Spike mutations to generate high-affinity double mutant complexes, suggesting variation in individual susceptibility to infection. Applying structural analysis to highly transmissible variants, we find that circulating point mutations S477N, E484K and N501Y form high-affinity complexes (~40% more than wild-type). By combining predicted affinities and available antibody escape data, we show that fast-spreading viral variants exploit combinatorial mutations possessing both enhanced affinity and antibody resistance, including S477N/E484K, E484K/N501Y and K417T/E484K/N501Y. Thus, three-dimensional modeling of the Spike/hACE2 complex predicts changes in structure and binding affinity that correlate with transmissibility and therefore can help inform future intervention strategies.