Tissue engineering has expanded into a highly versatile manufacturing landscape that holds great promise for advancing cardiovascular regenerative medicine. In this review, we provide a summary of the current state-of-the-art bioengineering technologies used to create functional cardiac tissues for a variety of applications in vitro and in vivo. Studies over the past few years have made a strong case that tissue engineering is one of the major driving forces behind the accelerating fields of patient-specific regenerative medicine, precision medicine, compound screening, and disease modeling. To date, a variety of approaches have been used to bioengineer functional cardiac constructs, including biomaterial-based, cell-based, and hybrid (using cells and biomaterials) approaches. While some major progress has been made using cellular approaches, with multiple ongoing clinical trials, cell-free cardiac tissue engineering approaches have also accomplished multiple breakthroughs, although drawbacks remain. This review summarizes the most promising methods that have been employed to generate cardiovascular tissue constructs for basic science or clinical applications. Further, we outline the strengths and challenges that are inherent to this field as a whole and for each highlighted technology.