This paper provides a review of recent developments in research and design practice surrounding the structural use of stainless steel, with an emphasis on structural stability. The nonlinear stress-strain characteristics of stainless steel, which are discussed first, give rise to a structural response that differs somewhat from that of structural carbon steel. Depending on the type and proportions of the structural element or system, the nonlinear material response can lead to either a reduced or enhanced capacity relative to an equivalent component featuring an elastic, perfectly plastic material response. In general, in strength governed scenarios, such as the in-plane bending of stocky beams, the substantial strain hardening of stainless steel gives rise to capacity benefits, while in stability governed scenarios, the early onset of stiffness degradation results in reduced capacity. This behaviour is observed at all levels of structural response including at cross-sectional level, member level and frame level, as described in the paper. Current and emerging design approaches that capture this response are also reviewed and evaluated. Lastly, with a view to the future, the application of advanced analysis to the design of stainless steel structures and the use of 3D printing for the construction of stainless steel structures are explored.