Low-rank matrix recovery problems are inverse problems which naturally arise in various fields like signal processing, imaging and machine learning. They are non-convex and NP-hard in full generality. It is therefore a delicate problem to design efficient recovery algorithms and to provide rigorous theoretical insights on the behavior of these algorithms. The goal of these notes is to review recent progress in this direction for the class of so-called "non-convex algorithms", with a particular focus on the proof techniques. Although they aim at presenting very recent research works, these notes have been written with the intent to be, as much as possible, accessible to non-specialists. These notes were written for an eight-hour lecture at Collège de France. The original version, in French, is available online 1 and the videos of the lecture can be found on the Collège de France website 2. The beginning takes inspiration from the review articles [Davenport and Romberg, 2016] and [Chen and Chi, 2018].