Valvular heart disease and congenital heart defects represent a major cause of death around the globe. Although current therapy strategies have rapidly evolved over the decades and are nowadays safe, effective, and applicable to many affected patients, the currently used artificial prostheses are still suboptimal. They do not promote regeneration, physiological remodeling, or growth (particularly important aspects for children) as their native counterparts. This results in the continuous degeneration and subsequent failure of these prostheses which is often associated with an increased morbidity and mortality as well as the need for multiple re-interventions. To overcome this problem, the concept of tissue engineering (TE) has been repeatedly suggested as a potential technology to enable native-like cardiovascular replacements with regenerative and growth capacities, suitable for young adults and children. However, despite promising data from pre-clinical and first clinical pilot trials, the translation and clinical relevance of such TE technologies is still very limited. The reasons that currently limit broad clinical adoption are multifaceted and comprise of scientific, clinical, logistical, technical, and regulatory challenges which need to be overcome. The aim of this review is to provide an overview about the translational problems and challenges in current TE approaches. It further suggests directions and potential solutions on how these issues may be efficiently addressed in the future to accelerate clinical translation. In addition, a particular focus is put on the current regulatory guidelines and the associated challenges for these promising TE technologies.