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Quantum effects play an essential role in modern cosmology. Perhaps the most striking example comes from large-scale structures, generally assumed to originate from vacuum quantum fluctuations and stretched by an expansion phase. Inflation is the leading paradigm in explaining this process. The various observational successes of inflationary models drive the scientific community into elaborating more and more stringent tests, which can simultaneously be used to probe beyond the simple slow-roll, single field inflation. However, inflation is not a theory, and going beyond inflation is a necessity. Various alternatives and/or complementary mechanisms to inflation have been invoked in the literature. The best-known cosmological models endowed with the capacity of explaining large-scale observations while avoiding the singularity form a class called non-singular bouncing models. The main features of these models are the presence of a contraction phase before expansion, and a never-vanishing scale factor. A non-singular bounce generally appears when quantum effects are part of the model, playing the role of a regulator leading to the avoidance of singularities. This thesis focuses on a Hamiltonian formulation of quantum effects in cosmology. We first explore stochastic perturbations in a collapsing universe. Then, we show that quantum cosmology with Bohmian mechanics resolves the initial singularity. Adding a non-minimal coupling of gravity with electromagnetism, we show that the generation of magnetic fields is possible. Finally, we apply the affine quantisation on the Brans-Dicke Theory, the prototype of modified gravity theories, and we discuss the quantum equivalence of the Jordan and Einstein frames within this framework. We show that in both frames a smooth bounce is expected, and that equivalence between frames holds at the quantum level.