In the past couple of decades, non-smooth convex optimization has emerged as a powerful tool for the recovery of structured signals (sparse, low rank, finite constellation, etc.) from possibly noisy measurements in a variety of applications in statistics, signal processing and machine learning. While the algorithms (basis pursuit, LASSO, etc.) are often fairly well established, rigorous frameworks for the exact analysis of the performance of such methods are only just emerging. The talk will introduce and describe a fairly general theory for how to determine the performance (minimum number of measurements, mean-square-error, probability-of-error, etc.) of such methods for various measurement ensembles (Gaussian, Haar, etc.). The framework enables one to assess the performance of these methods before actual implementation and allows one to optimally choose parameters such as regularizer coefficients, number of measurements, etc. The theory subsumes earlier results as special cases. It builds on an inconspicuous 1962 lemma of Slepian (for comparing Gaussian processes), as well as on a non-trivial generalization due to Gordon in 1988, and produces concepts from convex geometry (such as Gaussian widths and Moreau envelopes) in a very natural way. The talk will also consider extensions to certain non-Gaussian settings and their applications in massive MIMO, one-bit compressed sensing, graphical LASSO, and phase retrieval.