Matrix-product states have become the de facto standard for the representation of one-dimensional quantum many body states. During the last few years, numerous new methods have been introduced to evaluate the time evolution of a matrix-product state. Here, we will review and summarize the recent work on this topic as applied to finite quantum systems. We will explain and compare the different methods available to construct a time-evolved matrix-product state, namely the time-evolving block decimation, the MPO WI,II method, the global Krylov method, the local Krylov method and the one- and two-site time-dependent variational principle. We will also apply these methods to four different representative examples of current problem settings in condensed matter physics. •Introductory review of time-evolution methods for matrix-product states.•Detailed description of five state-of-the-art algorithms.•Extended comparison of numerical examples.•Extended error analysis of each algorithm.•Pseudo-code examples towards an implementation where necessary.