The semiclassical approximation in Quantum Field Theory is commonly used to study the propagation of quantized fields in classical backgrounds and accounts for fascinating non-perturbative quantum phenomena such as the spontaneous creation of particles induced by (non-trivial) external configurations. In this context, the computation of vacuum expectation values of physical observables becomes a complex issue, and advanced renormalization techniques are required to tame the new ultraviolet divergences caused by the external backgrounds. In curved spacetimes, all these interesting features are commonly studied within the Quantum Field Theory in Curved Spacetime framework, initiated in the early 60s and developed until nowadays. In semiclassical electrodynamics, different analytical methods, usually expressed in the modern language of Quantum Electrodynamics (QED), are employed to account for relevant non-perturbative quantum effects. This work aims to explore the interconnections between these two approaches, describing the underlying physics behind the semiclassical theory in a unified way. We analyze the intertwining relation between particle creation and quantum anomalies and study the backreaction problem in two-dimensional electrodynamics, investigating the range of validity of the semiclassical approach in a self-consistent way. We also work with different asymptotic expansions for the heat-kernel and the effective action in curved spacetimes and QED, focusing on its non-linear behavior. For the one-loop QED effective action, we find a new, resumed, asymptotic expansion that encapsulates in a non-perturbative factor all terms containing the field-strength invariants.