In these lectures, I will give concrete descriptions of categories of torsors under various classical reductive groups. The emphasis will be on working over a general base S rather than over a field. I will also explain how these torsors are mapped to each other using well-known exact sequences of algebraic groups between simply connected forms, adjoint forms, etc. The framework of Giraud's "Cohomologie non abélienne" will be used, but I will try to keep everything elementary, so that someone who is not familiar with stacks, gerbes, etc. should get a first idea of these concepts, without being lost in their generality.