The ability to create complex three-dimensional cellular models that can effectively replicate the structure and function of human organs and tissues in vitro has the potential to revolutionize medicine. Such models could facilitate the interrogation of developmental and disease processes underpinning fundamental discovery science, vastly accelerate drug development and screening, or even be used to create tissues for implantation into the body. Realization of this potential, however, requires the recreation of complex biochemical, biophysical, and cellular patterns of 3D tissues and remains a key challenge in the field. Recent advances are being driven by improved knowledge of tissue morphogenesis and architecture and technological developments in bioengineering and materials science that can create the multidimensional and dynamic systems required to produce complex tissue microenvironments. In this article, we discuss challenges for in vitro models of tissues and organs and summarize the current state-of-the art in biomaterials and bioengineered systems that aim to address these challenges. This includes both top-down technologies, such as 3D photopatterning, magnetism, acoustic forces, and cell origami, as well as bottom-up patterning using 3D bioprinting, microfluidics, cell sheet technology, or composite scaffolds. We illustrate the varying ways that these can be applied to suit the needs of different tissues and applications by focussing on specific examples of patterning the bone-tendon interface, kidney organoids, and brain cancer models. Finally, we discuss the challenges and future prospects in applying materials science and bioengineering to develop high-quality 3D tissue structures for in vitro studies.