Biodegradable terephthalate polymers are described comprising the recurring monomeric units shown in formula I: wherein R is a divalent organic moiety; R′ is an aliphatic, aromatic or heterocyclic residue; x is ≧1; and n is 0-5,000, wherein the biodegradable polymer is biocompatible before and upon biodegradation. Processes for preparing the polymers, compositions containing the polymers and biologically active substances, articles useful for implantation or injection into the body fabricated from the compositions, and methods for controllably releasing biologically active substances using the polymers, are also described.