Reheating after inflation occurs due to particle production by the oscillating inflaton field. In this paper we briefly describe the perturbative approach to reheating, and then concentrate on effects beyond the perturbation theory. They are related to the stage of parametric resonance, which we call preheating. It may occur in an expanding universe if the initial amplitude of oscillations of the inflaton field is large enough. We investigate a simple model of a massive inflaton field $\phi$ coupled to another scalar field $\chi$ with the interaction term $g^2{\phi }^2{\chi }^2$. Parametric resonance in this model is very broad. It occurs in a very unusual stochastic manner, which is quite different from parametric resonance in the case when the expansion of the universe is neglected. Quantum fields interacting with the oscillating inflaton field experience a series of kicks which, because of the rapid expansion of the universe, occur with phases uncorrelated to each other. Despite the stochastic nature of the process, it leads to exponential growth of fluctuations of the field $\chi$. We call this process stochastic resonance. We develop the theory of preheating taking into account the expansion of the universe and back reaction of produced particles, including the effects of rescattering. This investigation extends our previous study of reheating after inflation. We show that the contribution of the produced particles to the effective potential $V\left(\phi \right)$ is proportional not to ${\phi }^2$, as is usually the case, but to $\left|\phi \right|$. The process of preheating can be divided into several distinct stages. In the first stage the back reaction of created particles is not important. In the second stage back reaction increases the frequency of oscillations of the inflaton field, which makes the process even more efficient than before. Then the effects related to scattering of $\chi$ particles on the oscillating inflaton field terminate the resonance. We calculate the number density of particles $n_{\chi }$ produced during preheating and their quantum fluctuations $〈{\chi }^2〉$ with all back reaction effects taken into account. This allows us to find the range of masses and coupling constants for which one can have efficient preheating. In particular, under certain conditions this process may produce particles with a mass much greater than the mass of the inflaton field.

The theory of reheating of the Universe after inflation is developed. We have found that typically at the first stage of reheating the classical inflation field $\phi$ rapidly decays into $\phi$ particles or into other bosons due to a broad parametric resonance. Then these bosons decay into other particles, which eventually become thermalized. Complete reheating is possible only in those theories where a single particle $\phi$ can decay into other particles. This imposes strong constraints on the structure of inflationary models, and implies that the inflation field can be a dark matter candidate.

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