Mechanical signals are increasingly recognized as overarching regulators of cell behaviour, controlling stemness, organoid biology, tissue development and regeneration. Moreover, aberrant mechanotransduction is a driver of disease, including cancer, fibrosis and cardiovascular defects. A central question remains how cells compute a host of biomechanical signals into meaningful biological behaviours. Biomaterials and microfabrication technologies are essential to address this issue. Here we review a large body of evidence that connects diverse biomaterial-based systems to the functions of YAP/TAZ, two highly related mechanosensitive transcriptional regulators. YAP/TAZ orchestrate the response to a suite of engineered microenviroments, emerging as a universal control system for cells in two and three dimensions, in static or dynamic fashions, over a range of elastic and viscoelastic stimuli, from solid to fluid states. This approach may guide the rational design of technological and material-based platforms with dramatically improved functionalities and inform the generation of new biomaterials for regenerative medicine applications.