Crosslinked polymer compositions have backbones with first and second divalent saturated aliphatic moieties, a divalent saturated aliphatic secondary alcohol moiety, and a trivalent saturated aliphatic moiety. Hydrolytically labile ester bonds joined together these moieties. These polyesters may be polycondensation reaction products of a diol, a triol and a diacid. A molar ratio of the first divalent saturated aliphatic moiety, the divalent saturated aliphatic secondary alcohol moiety, and the trivalent saturated aliphatic moiety to the second divalent saturated aliphatic moiety is in the range of about 0.85 to about 1.5. Preferably, these polyesters are non-cytotoxic, biocompatible, bioabsorbable, or exhibit shape memory behavior with at least one transition temperature of greater than about 30.degree. C. and less than about 100.degree. C. and most preferably exhibit each of these qualities. The compositions may be adapted for a wide variety of uses, including medical applications.