Shape memory polymers (SMP) belong to the class of stimuli-responsive materials and have generated significant research interest. Their capability to retain an imposed, temporary shape and to recover the initial, permanent shape upon exposure to an external stimulus depends on the "functional determinants", which in simplistic terms, can be divided into structural/morphological and processing/environmental factors. The primary aim of the first part of this review is to reflect the knowledge about these fundamental relationships. In a next step, recent advances in shape memory polymer composites are summarized. In contrast to earlier reviews, studies on the impairment of shape memory properties through various factors, such as aging, compression and hibernation, lubricants, UV light and thermo-mechanical cycling, are extensively reviewed. Apart from summarizing the state-of-the-art in SMP research, recent progress is commented.