Polymer-based biomaterials are extensively used in all disciplines of clinical medicine and innovations in biomaterial science are building a product pipeline, e.g., of future cardiovascular implants. Still, cardiovascular applications demand a number of extensive requirements of properties and functions to be fulfilled by the polymer matrix. This report provides an overview on some of these issues and how they can be addressed by a tailored design of novel polymer-based biomaterials. Multifunctional shape-memory polymers are highlighted as a class of materials that combine biocompatibility and the capability for stimuli-induced active movements for anchoring of implants with a controlled degradation and drug release profile to enable a functional regeneration of the tissue at the application site.