Tissue engineering has been recognized as a translational approach to replace damaged tissue or whole organs. Engineering tissue, however, faces an outstanding knowledge gap in the challenge to fully recapitulate complex organ-specific features. Major components, such as cells, matrix, and architecture, must each be carefully controlled to engineer tissue-specific structure and function that mimics what is found in vivo. Here we review different methods to engineer tissue, and discuss critical challenges in recapitulating the unique features and functional units in four major organs-the kidney, liver, heart, and lung, which are also the top four candidates for organ transplantation in the USA. We highlight advances in tissue engineering approaches to enable the regeneration of complex tissue and organ substitutes, and provide tissue-specific models for drug testing and disease modeling. We discuss the current challenges and future perspectives toward engineering human tissue models.