These lecture notes present an introduction to large deviations in stochastic dynamical systems, adapted from a lecture series given at the Les Houches school ``Large Deviations: Theory and Applications'' in July 2024. The exposition develops the theory through a unified methodology: WKB analysis of backward Kolmogorov equations, which reveals the Hamiltonian structure underlying all large deviation principles considered here. We treat three asymptotic regimes systematically. First, the small-noise limit for stochastic differential equations and the large-volume limit for Markov jump processes, where the Large Deviation Principle emerges from the Law of Large Numbers through progressively finer characterizations of fluctuations. Second, the long-time limit combined with small noise, yielding the quasipotential formalism for metastability and transition rates in both equilibrium and nonequlibrium systems. Third, the long-time limit at finite noise, connecting to spectral methods via tilted generators and enabling reduction of multiscale systems.We survey computational approaches including minimum action methods, their geometric variants for quasipotential calculations, and neural network techniques. The Schlögl model of chemical bistability serves as a running example, illustrating both scalar and spatially extended settings. Appendices provide self-contained background on stochastic calculus and Markov jump processes.