The restoration or de novo engineering of heart structures poses a challenge because of the unique structure and physical properties of the heart. The heart is a heterogeneous, complex helical structure with asymmetric and anisotropic features. Hence, it is variably built and consists of spiraling muscle bands (first described by Torrent-Guasp), valves, coronary vessels, and a conduction system. Therefore, successful construction of muscular or valvular grafts needs to meet specific prerequisites, such as mechanical stability, optimal porosity, and contractile function. A series of studies have reported on various mixtures of scaffolds and (stem) cells and subsequent production of spontaneously contractile cardiac grafts in vitro. In vivo studies have provided evidence of engraftment of such bioartificial myocardial grafts and of improved heart function of the host. Two fundamental types of bioartificial/engineered valves have been reported: decellularized xenogeneic valves and recellularized valves with the recipient's own cells. Large-scale clinical results are awaited. Finally, bioartificial vessels are being produced, either through de novo construction from collagens and cells or from previously harvested recipient's own fibroblasts and endothelial cells. The main goal envisioned here is long-term patency following implantation in vivo. This review surveys upon recent developments and indicates caveats in the field of tissue engineering of cardiac, valvular, and vascular structures.