Path integrals in curved space and the worldline formalism

We describe how to construct and compute unambiguously path integrals for particles moving in a curved space, and how these path integrals can be used to calculate Feynman graphs and effective actions for various quantum field theories with external gravity in the framework of the worldline formalism. In particular, we review a recent application of this worldline approach and discuss vector and antisymmetric tensor fields coupled to gravity. This requires the construction of a path integral for the $N=2$ spinning particle, which is used to compute the first three Seeley--DeWitt coefficients for all $p$-form gauge fields in all dimensions and to derive exact duality relations.