As the use of soft and compliant materials is rapidly penetrating modern engineering practice, materials are expected to perform at large strains and deep into the nonlinear regime. Although pushing materials to their extremes often exposes unique phenomena and instabilities that were traditionally avoided in engineering, an emerging field of mechanics aims to harness and exploit these rapid morphological changes for design of novel devices. Remarkably, nature has also been shown to exploit instability; a bi-stable mechanism is thought to absorb shocks in our muscles, and growth-induced instability is thought to promote rapid changes of shape and even cell movement. A ubiquitous growth mechanism, in both natural and engineered systems, is surface growth, in which material associates or dissociates on the boundaries of a body. It is the fundamental mechanism by which biological materials grow and is increasingly applied in engineering processes for fabrication and self-assembly. In this talk I will overview some recent work which explores the nonlinear mechanics of growth and instability in soft and biological materials. We will discuss multiple instability modes that appear in finite elastic materials, then we will shift to discuss recent progress in modeling surface growth processes with coupled diffusion.