We analyze Runge–Kutta (RK) implicit-explicit (IMEX) time-stepping schemes to approximate the Cauchy problem associated with a partial differential equation of convection-diffusion-type, i.e., comprising a first-order part (the transport operator) and a second-order part (the diffusion operator). The proposed approach departs from the traditional method of lines, as the transport part is discretized in space using continuous finite elements with streamline upwind Petrov–Galerkin (SUPG) stabilization, whereas the diffusion part is discretized using the plain Galerkin method. Contraction in a suitable norm composed of the L^2-norm augmented with a term accounting for the SUPG stabilization is proved for the first-order Lie splitting scheme, the second-order Strang splitting scheme, and two third-order Runge–Kutta IMEX methods. All these results require that the coefficient weighting the SUPG stabilization and the Courant number are small enough, uniformly with respect to the local Peclet number. The contraction property of the third-order IMEX methods also assumes that the discrete diffusion and transport operators commute. This is joint work with Jean-Luc Guermond (Texas A&M).