Described are shape-memory polymers that have a composite prepolymer crosslinked with a stoichiometric amount of a multifunctional crosslinker, the composite prepolymer having a branched or telechelic prepolymer having a low polydispersity reacted with a non-crystalline chain extender. Also described are methods of making shape-memory polymers by reacting a branched or telechelic prepolymer having a low polydispersity with a non-crystalline chain extender to form a composite prepolymer, and crosslinking a stoichiometric amount of a multifunctional crosslinker with the composite prepolymer, thereby forming the shape-memory polymer.