The ability of polymers to respond to external stimuli is of high scientific and technological significance. In the last few years, research activities have been intensified substantially, exploring whether stimuli-sensitive polymers can be designed that move actively. In this review actively-moving materials were classified according to the underlying mechanisms enabling the shape changes: shape-memory polymers and shape-changing polymers/shape-changing gels were identified. The application spectra of these materials as well as the current developments were elucidated and general molecular design principles presented. When applicable, a further distinction according to the applied stimulus was made.