This thesis consists of three chapters. The first of these surveys the field of nonparametric inference under shape constraints, focussing in particular on the topics of shape-restricted regression and shape-constrained density estimation. In the second chapter, we investigate the adaptation properties of the log-concave maximum likelihood estimator of a multivariate log-concave density. Our main theoretical results demonstrate that in certain situations where the true density has additional structure, the estimator can attain rates of convergence (with respect to squared Hellinger distance or Kullback-Leibler divergence) that are strictly faster than the global minimax convergence rate. We illustrate three different types of adaptive behaviour in dimensions 2 and 3 through sharp oracle inequalities. Our approach entails developing local bracketing entropy bounds for Hellinger neighbourhoods of log-concave densities that belong to the special subclasses described above. To this end, we apply techniques from convex geometry and real analysis to elucidate the structural properties of such densities, and obtain some results of independent interest. In the third chapter, we consider the nonparametric estimation of an S-shaped regression function. The least squares estimator provides a very natural, tuning-free approach, but results in a non-convex optimisation problem, since the inflection point is unknown. We show that the estimator may nevertheless be regarded as a projection onto a finite union of convex cones, which allows us to propose a mixed primal-dual bases algorithm for its efficient, sequential computation. After developing a general projection framework that demonstrates the consistency and robustness to misspecification of the estimator, we prove worst-case and adaptive risk bounds for the estimation of the regression function, in the form of sharp oracle inequalities, and establish bounds on the rate of convergence of the estimated inflection point. These theoretical results reveal not only that the estimator achieves the minimax optimal rate for both the estimation of the regression function and its inflection point (up to a logarithmic factor in the latter case), but also that it is able to achieve an almost-parametric rate when the true regression function is piecewise affine with not too many affine pieces. Simulations also confirm the desirable finite-sample properties of the estimator, and our algorithm is implemented in the R package 'Sshaped'.