Polymer smart materials are a broad class of polymeric materials that can change their shapes, mechanical responses, light transmissions, controlled releases, and other functional properties under external stimuli. A good understanding of the aspects controlling various types of shape memory phenomena in shape memory polymers (SMPs), such as polymer structure, stimulus effect and many others, is not only important for the preparation of new SMPs with improved performance, but is also useful for the optimization of the current ones to expand their application field. In the present era, simple understanding of the activation mechanisms, the polymer structure, the effect of the modification of the polymer structure on the activation process using fillers or solvents to develop new reliable SMPs with improved properties, long lifetime, fast response, and the ability to apply them under hard conditions in any environment, is considered to be an important topic. Moreover, good understanding of the activation mechanism of the two-way shape memory effect in SMPs for semi-crystalline polymers and liquid crystalline elastomers is the main key required for future investigations. In this article, the principles of the three basic types of external stimuli (heat, chemicals, light) and their key parameters that affect the efficiency of the SMPs are reviewed in addition to several prospective applications.