Rigorous statistical mechanics deals with stochastic systems that have a large number of components and for which geometry often plays an important role. The main goal is to understand their average behavior and deviations from that. Statistical mechanics has many applications to physics and other fields, but in this talk I will only focus on some aspects of its mathematical theory which combine discrete probability, stochastic processes and complex analysis. For concreteness, I will discuss two specific (two-dimensional) models, percolation and the Ising model, both with a long history. After introducing the two models, I will present an approach to the study of two-dimensional systems that leads to a special family of random fractal curves and that has produced, in the last twenty years, deep results and major breakthroughs.