Shape memory (SM) biodegradable polymers (SMPs) and related composites are emerging smart materials in different applications. SMPs may adopt one (dual-shape), two (triple-shape) or several (multishape) stable temporary shapes and recover their permanent shape (or other temporary shapes in case of multi-shape versions) upon the action of an external stimulus. The external stimulus may be temperature, pH, water, light irradiation, redox condition etc. In most cases, however, the SMPs are thermally activated. The 'switching' or transformation temperature (T trans ), enabling the material to return to its permanent shape, is either linked with the glass transition (T g ) or with the melting temperature (T m ).Thus, SMPs are often subdivided based on their switch types into T g -or T m -based SMPs. As reversible 'switches' other mechanisms such as liquid crystallization and related transitions, supermolecular assembly/disassembly, irradiation-induced reversible network formation, formation and disruption of a percolation network, may also serve [1]. The permanent shape is guaranteed by physical or chemical network structures. The latter may be both on molecular and supramolecular levels. Linkages of these networks are termed net points. The temporary shape is created by mechanical deformation above T trans . In some cases the deformation temperature may be below Abstract. Shape memory polymers (SMPs) are capable of memorizing one or more temporary shapes and recovering to the permanent shape upon an external stimulus that is usually heat. Biodegradable polymers are an emerging family within the SMPs. This minireview delivers an overlook on actual concepts of molecular and supramolecular architectures which are followed to tailor the shape memory (SM) properties of biodegradable polyesters. Because the underlying switching mechanisms of SM actions is either related to the glass transition (T g ) or melting temperatures (T m ), the related SMPs are classified as T g -or T m -activated ones. For fixing of the permanent shape various physical and chemical networks serve, which were also introduced and discussed. Beside of the structure developments in one-way, also those in two-way SM polyesters were considered. Adjustment of the switching temperature to that of the human body, acceleration of the shape recovery, enhancement of the recovery stress, controlled degradation, and recycling aspects were concluded as main targets for the future development of SM systems with biodegradable polyesters.