The Cremona group is the group of birational transformations of the projective space. In dimension 2 the Cremona group was already studied intensely in the 19th century, but many classical problems have only been solved in the last decades. In higher dimensions, most questions remain open. In a series of 3 lectures I will discuss various questions, examples and results about Cremona groups, such as the classification of its simple subgroups in dimension 2, and explain some of the underlying techniques. Each lecture will be self-contained.