Quantum fields in curved spacetime undergo fluctuations that produce non-vanishing vacuum expectation values of the stress-energy tensor, i.e., energy can be generated due to the gravitational field. The same happens for other type of background fields like gauge or scalars. This effect plays an important role in the early Universe, in astrophysical compact objects, and in strong electromagnetic phenomena. However, the computation of the stress-energy tensor, among others, is a highly nontrivial issue. In particular, non-trivial divergences appear when computing expectation values of local observables. The objective of my thesis is to tackle this issue by studying regularization and renormalization mechanisms for quantum fields in curved spacetime, especially in Friedman-Robertson-Walker-Lemaitre spacetimes. On the one hand, this will be done by extending adiabatic regularization to include interacting fields (scalar, gauge fields). On the other hand, running of the coupling constant by introducing a mass parameter will be computed for general curved spacetime and a subtraction scheme, that naturally incorporates decoupling for higher massive fields will be obtained. A particular application will be given in the context of the cosmological constant problem.