© 2019 American Mathematical Society. An initial-boundary value problem with a Caputo time derivative of fractional order ¿ ¿ (0, 1) is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple framework for the analysis of the error of L1-type discretizations on graded and uniform temporal meshes in the L¿ and L2 norms. This framework is employed in the analysis of both finite difference and finite element spatial discretiztions. Our theoretical findings are illustrated by numerical experiments.