The human heart lacks significant regenerative capacity; thus, the solution to heart failure (HF) remains organ donation, requiring surgery and immunosuppression. The demand for constructed cardiac tissues (CCTs) to model and treat disease continues to grow. Recent advances in induced pluripotent stem cell (iPSC) manipulation, CRISPR gene editing, and 3D tissue culture have enabled a boom in iPSC-derived CCTs (iPSC-CCTs) with diverse cell types and architecture. Compared with 2D-cultured cells, iPSC-CCTs better recapitulate heart biology, demonstrating the potential to advance organ modeling, drug discovery, and regenerative medicine, though iPSC-CCTs could benefit from better methods to faithfully mimic heart physiology and electrophysiology. Here, we summarize advances in iPSC-CCTs and future developments in the vascularization, immunization, and maturation of iPSC-CCTs for study and therapy.