Variants of SARS-CoV-2 have emerged which contain multiple substitutions in the surface spike glycoprotein that have been associated with increased transmission and resistance to neutralising antibodies and antisera. We have examined the structure and receptor binding properties of spike proteins from the B.1.1.7 (UK) and B.1.351 (SA) variants to better understand the evolution of the virus in humans. Both variants’ spikes have the same mutation, N501Y, in their receptor-binding domains that confers tighter ACE2 binding and this substitution relies on a common earlier substitution (D614G) to achieve the tighter binding. Each variant spike has also acquired a key change in structure that impacts virus pathogenesis. Unlike other SARS-CoV-2 spikes, the spike from the UK variant is stable against detrimerisation on binding ACE2. This feature primarily arises from the acquisition of a substitution at the S1-S2 furin site that allows for near-complete cleavage. In the SA variant spike, the presence of a new substitution, K417N, again on the background of the D614G substitution, enables the spike trimer to adopt fully open conformations that are required for receptor binding. Both types of structural change likely contribute to the increased effectiveness of these viruses for infecting human cells.