Gradient-based optimization is a potent tool in many design processes today. It is particularly useful in industries where weight considerations are crucial, such as aerospace, but can also be exploited in for example civil engineering applications to reduce the material use and thereby the environmental impact. With the advent of advanced manufacture methods, it even possesses the potential to design novel materials with enhanced properties that naturally occurring materials lack. Unfortunately, most research on the subject often limits itself to linear problems, wherefore the optimization's utility in solving intricate non-linear problems is still comparatively rudimentary. The aim of this thesis is therefore to investigate gradient-based optimization of various non-linear structural problems, while addressing their inherent numerical and modeling complexities. This thesis contains an introduction to gradient-based optimization of non-linear structures and materials, involving both shape and topology optimization. To start, the governing equations of the macroscopic and microscopic problems are described. A multi-scale framework which details the transition between the scales is defined. A substantial part of the thesis is dedicated to eigenvalue problems in topology optimization, and the numerical issues that they accompany. Specifically, the effects of finite deformations on the topology optimized design taking into account eigenfrequencies, structural stability or elastic wave propagation are scrutinized. A fictitious domain approach to topology optimization is employed, wherein void regions are modeled via an ersatz material with low stiffness. Unfortunately, this brings about artificial eigenmodes and convergence problems in the finite element analyzes. Two methods which deal with both of the aforementioned problems are proposed, and their efficacy is illustrated via several numerical examples. The use of shape optimization to post-process topology optimized designs is investigated for problems where accurate boundary descriptions are crucial to capture the physics, as is the case in contact problems. To take this concept further, a simultaneous topology and shape optimization method is proposed, which allows parts of the structural boundaries to be modeled exactly up to numerical precision. This approach is proven to be especially useful in the design of pressure-driven soft robots.