Linear periodic systems originate in various control fields involving periodic phenomena. In the beginnings of algorithmic developments for periodic systems, computational detours have been often employed to reduce the computational problems for periodic systems to those for standard systems. Well-known techniques are the employment of lifted representations in discrete-time or the use of periodic generators in continuous-time. New computational paradigms evolved later, whose main ingredients are numerically reliable and efficient algorithms for manipulating matrix products without forming explicitly them, reduction of large-scale structured matrix pencils without building the underlying pencils, computing with discrete-time periodic system models with time-varying state dimensions, or solving periodic matrix differential equations using multiple shooting techniques. The paper will give a succinct overview of all these new developments and will point out some of still existing open computational problems.