The propagation of radiation in complex material geometries is governed by an integro-differential equation known as the transport equation. Although several analytical solutions are known, these apply for simple radiation sources, for a limited number of interaction mechanisms (often just one or two), and for extremely simplified geometries (typically infinite media). Instead, the Monte Carlo (MC) method provides an efficient approach to solve the transport equation for nearly arbitrary radiation sources and complex material geometries. The MC method consists in numerically simulating an ensemble of particle trajectories emanating from the radiation source and undergoing prescribed interaction mechanisms, each governed by its interaction cross section (indicating the likelihood of each process) and by a differential cross section (describing how various possible final states are populated as a result of each interaction). The generation of secondary particles from inelastic interactions is naturally included, thus making the MC method a natural framework to simulate coupled hadronic and electromagnetic radiation showers. A statistical analysis of the simulated ensemble of particle trajectories allows for an estimate of a broad range of radiometric observables, e.g. energy deposition, particle production spectra, radioactive inventories, etc. In this talk, the fundamental ideas underlying the MC method for radiation transport problems shall be discussed, paying particular attention to highlight not only its virtues but also its assumptions and limitations.