The talk will discuss recent advances in the analysis of non-asymptotic estimation error and structured statistical recovery based on norm regularized regression, such as Lasso, using geometric techniques. Analysis of estimation error for regularized problems needs to consider four aspects: the norm, the loss function, the design matrix, and the noise model. The talk will discuss new results on all four aspects. In particular, the new results are applicable to any norm, general sub-Gaussian design matrices, including anisotropic designs, and general convex loss functions, including least squares and generalized linear models. Gaussian width, as a measure of size of sets, and associated geometric tools, especially generic chaining, play a key role in our analysis.