In this article we review the microfabrication approaches, with a focus on bioprinting and organ-on-chip technologies, used to engineer cardiac tissue. First, we give a brief introduction to heart anatomy and physiology, and the developmental stages of the heart from fetal stages to adulthood. We also give information on the cardiac tissue microenvironment, including the cells residing in the heart, the biochemical composition and structural organization of the heart extracellular matrix, the signaling factors playing roles in heart development and maturation, and their interactions with one another. We then give a brief summary of both cardiovascular diseases and the current treatment methods used in the clinic to treat these diseases. Second, we explain how tissue engineering recapitulates the development and maturation of the normal or diseased heart microenvironment by spatially and temporally incorporating cultured cells, biomaterials, and growth factors (GF). We briefly expand on the cells, biomaterials, and GFs used to engineer the heart, and the limitations of their use. Next, we review the state-of-the-art tissue engineering approaches, with a special focus on bioprinting and heart-on-chip technologies, intended to (i) treat or replace the injured cardiac tissue, and (ii) create cardiac disease models to study the basic biology of heart diseases, develop drugs against these diseases, and create diagnostic tools to detect heart diseases. Third, we discuss the recent trends in cardiac tissue engineering, including the use of machine learning, CRISPR/Cas editing, exosomes and microRNAs, and immune modeling in engineering the heart. Finally, we conclude our article with a brief discussion on the limitations of cardiac tissue engineering and our suggestions to engineer more reliable and clinically relevant cardiac tissues.