A model category is a category with some extra structure which makes it possible to do homotopy theory. As it turns out there are lots of examples, and not all ofthem are topological in nature. The aim of the project is to learn about modelcategories and Quillen adjunctions, which are the appropriate \maps" betweenthem. In this framework one can give a conceptual deffnition of derived functors,such as T or and Ext.We can then study one or more examples in further detail. The most important example is probably the Quillen equivalence between topological spaces and sim-plicial sets. Another interesting example is the various model structures on chaincomplexes over a ring.