This thesis is composed of two parts. The first part is devoted to inference for discretely observed diffusion processes. We first describe the general methodology of martingale estimating functions and show that this technique is particularly well suited to polynomial diffusions. We then use that theory as a building block to devise a new class of estimators for locally stationary diffusions. The second part of the thesis is concerned with problems related to robustness theory. We derive robust estimators and tests for randomly censored survival data, which arise when the time until a certain event occurs is not fully observable. Then, we study the general convex loss Lasso for the linear model in a high-dimensional setting. We present new bounds for the estimation and prediction errors, and show how these bounds depend on the choice of the loss function through a well-known term in robust statistics.