The interaction between the SARS-CoV-2 virus Spike protein receptor binding domain (RBD) and the ACE2 cell surface protein is required for viral infection of cells. Mutations in the RBD domain are present in SARS-CoV-2 variants of concern that have emerged independently worldwide. For example, the more transmissible B.1.1.7 lineage has a mutation (N501Y) in its Spike RBD domain that enhances binding to ACE2. There are also ACE2 alleles in humans with mutations in the RBD binding site. Here we perform a detailed affinity and kinetics analysis of the effect of five common RBD mutations (K417N, K417T, N501Y, E484K and S477N) and two common ACE2 mutations (S19P and K26R) on the RBD/ACE2 interaction. We analysed the effects of individual RBD mutations, and combinations found in new SARS-CoV-2 variants first identified in the UK (B.1.1.7), South Africa (B.1.351) and Brazil (P1). Most of these mutations increased the affinity of the RBD/ACE2 interaction. The exceptions were mutations K417N/T, which decreased the affinity. Taken together with other studies, our results suggest that the N501Y and S477N mutations primarily enhance transmission, the K417N/T mutations facilitate immune escape, and the E484K mutation facilitates both transmission and immune escape.