Shape memory polymers (SMPs) have attracted significant attention from both industrial and academic researchers, due to their useful and fascinating functionality. One of the most common and studied external stimuli for SMPs is temperature; other stimuli include electric fields, light, magnetic fields, water, and irradiation. Solutions for SMPs have also been extensively studied in the past decade. In this research, we review, consolidate, and report the major efforts and findings documented in the SMP literature, according to different external stimuli. The corresponding mechanisms, constitutive models, and properties (i.e., mechanical, electrical, optical, shape, etc.) of the SMPs in response to different stimulus methods are then reviewed. Next, this research presents and categorizes up-to-date studies on the application of SMPs in dynamic building structures and components. Following this, we discuss the need for studying SMPs in terms of kinetic building applications, especially about building energy saving purposes, and review recent two-way SMPs and their potential for use in such applications. This review covers a number of current advances in SMPs, with a view towards applications in kinetic building engineering.