In generality, homotopy theory is the study of mathematical contexts in which functions are equipped with a concept of homotopy between them. A key aspect of the theory is that the concept of isomorphism is relaxed to that of homotopy equivalence: Where a function is regarded as invertible if there is a reverse function such that both composites are equal to the identity, for a homotopy equivalence one only requires the composites to be homotopic to the identity. Model categories give rise to a large class of homotpy theories, and provide a setting that is suitable for concrete calculations. My aim is to explain the basics of category theory, define model categories and show how to work in this setting. I will state the Quillen transfer theorem and illustrate it in the theory of homological algebra. Given the time, I will say something about the derived critical locus in the derived algebraic geometry.