Abstract This paper presents a review of finite- and fixed-time convergent algorithms for dynamic systems of various dimensions and relative degrees, with or without disturbances. The review mainly focuses on finite- and fixed-time convergent algorithms, which provide explicit and easily computable upper estimates for the corresponding convergence times. The finite- and fixed-time convergent algorithms are classified according to their actions and structural properties: regulators, observers (differentiators), controllers; discontinuous and continuous, non-smooth and smooth, non-adaptive and adaptive. Some fixed-time convergent algorithms are illustrated by simulation examples displaying typical time histories of the state/estimate trajectories in each case.