We will discuss the main conjectures, problems and methods in mathematical general relativity. In the first lecture we will present the cosmic censorship conjectures and discuss various works in that direction. In particular, we will present useful methods such as the double null foliation and the short pulse method and explain how they can be used for important problems such as the formation of trapped surfaces and the study of the Cauchy horizon and of the role of regularity in the conjectures.