During the past 5-10 years, there has been considerable progress in the construction of renormalized, perturbative, interacting quantum field theories on general curved spacetime manifolds. This progress has become possible with the help of mathematical methods from “Microlocal Analysis” on the one hand, and a new conceptual viewpoint on perturbative QFT emphasizing the algebraic and “local and covariant” nature of quantum fields. The mathematical methods were important in order to control the singularities of field products, and the algebraic viewpoint was essential to formulate the theory without using a preferred “vacuum state”, which is not available in a general curved space. In my talk, I want to present the key ideas in the subject and outline its present status.