**Abstract**
The Feynman-Vernon approach to open quantum systems is to express the evolution of the reduced density matrix of the system as a double path integral, where one path ("forward path") comes from the unitary U acting from the left, and one path ("backward path") comes from the inverse unitary U^{\dagger} acting from the right. In the open systems context the Feynman-Vernon approach is closely related to the Keldysh theory, where the forward-backward paths correspond to the positive/negative parts of the Keldysh contour. I will first present an alternative derivation of Feynman-Vernon theory by analyzing the von-Neumann-Liouville equation as a linear evolution law in the space of Hermitian operators. Results on full counting statistics (generating functions of energy changes in one or several baths), that are somewhat complicated to obtain in the path integral language, then emerge in a much simpler way. I will then look at systems interacting linearly with baths that are not harmonic, but instead characterized by an expansion in cumulants. Every non-zero cumulant of certain environment correlation functions then gives a kernel in a higher-order term in the Feynman-Vernon action, and I will discuss a few of these higher-order terms. This talks is partly work in progress; results so far are presented in joint paper with Ryoichi Kawai and Ketan Goyal, available as arXiv:1907.02671.