In this review, we summarize the main ideas of perturbative algebraic quantum field theory, which is a rigorous framework combining some of the Haag-Kastler axioms with perturbative methods involving formal power series. It allows for the construction of interacting QFT models in four spacetime dimensions and works on arbitrary globally hyperbolic manifolds. This approach has also led to the development of a non-perturbative construction of local nets of C*-algebras for interacting theories, which will also be discussed at the end of this review.