It is the purpose of this paper to consider a selection of adapted numerical methods for the solution of evolutionary problems showing a periodic behaviour in their dynamics. The treatise will cover both evolutionary problems in time and space (i.e. reactiondiffusion problems generating a periodic wavefront in time and space) and problems with memory (i.e. Volterra integral equations with periodic solutions), also underlining the necessity to introduce suitable quadrature rules adapted to the problems under investigation. The approach is mainly problem-oriented, in order to match qualitative properties of the problem with the numerical methods, in order to get better accuracy, efficiency and stability properties with respect to existing numerical schemes.
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