2 Typically, the state of matter after preheating is highly non-thermal, and thus must be followed by a phase of thermalization. Cosmological inflation [1] is a phase of accelerated ex- pansion of space. In the context of General Relativity as the theory describing space and time, inflation re- quires scalar field matter. More precisely, the energy- momentum tensor of matter must be dominated by the almost constant potential energy density of the scalar field φ . A scalar field is postulated to exist in the SM of particle physics: the Higgs field used to give elementary fermions their masses. To serve as a Higgs field, its potential en- ergy must have a minimum at a non-trivial field value. The standard example is where η is the vacuum expectation value of φ . It is as- sumed that at high temperatures the symmetry is re- stored by finite temperature effects (see e.g. [12] for a review of field theory methods used in inflationary cos- mology) and φ = 0. Once the temperature T falls below a critical value T$_{c}$ , φ ceases to be trapped and will start to roll towards one of the lowest energy states φ = ± η . The SM Higgs must have a coupling constant λ which is set by the gauge coupling constant and cannot be suffi- ciently small to yield a long time period of slow rolling of φ which is required to obtain enough inflation (except possibly if φ is non-minimally coupled to gravity [13]). The first goal of this review article is to present an introduction to the theory of preheating after inflation. In Section 2 of this article, we give a lightning review of inflationary cosmology. Section 3 is the most important section of this review in which we present a comprehen- sive analysis of preheating. The efficiency of preheat- ing turns out to be rather model-dependent. We first discuss “standard preheating” which will typically occur in simple single-field inflation models like those used in “Chaotic Inflation”. The efficiency of preheating can be much higher if a tachyonic direction develops, which is what occurs in certain small field inflation models and in “Hybrid Inflation”, a model involving two scalar fields. It turns out that while preheating leads to a very rapid start to the process of energy transfer from the inflaton to SM matter, it typically does not drain most of the en- ergy of the inflaton field. This happens in a second stage, a stage characterized by the nonlinear interactions of the fluctuation modes which have been highly excited by the preheating process. According to recent studies [11], this process is turbulent. The initial stage of turbulence (af- ter which the bulk of the energy density is no longer in the inflaton field) is rapid, but the actual thermalization of the decay products takes much longer. These issues are briefly discussed in Section 4. Section 5 focuses on reheating in supersymmetric models. Finally, in Section 6 we give a brief overview of a number of applications of preheating in inflationary cosmology. Hence, scalar field-driven inflation requires us to go beyond the SM of particle physics. Once one makes this step, there are typically many candidate scalar fields which could be the inflaton, in particular in supersym- metric models. For cosmological studies, the precise nature of the in- flaton is often secondary and hence simple toy models are used. “New” inflation [14, 15] maintains the idea that φ begins trapped near φ = 0. Inflation takes place during the period when φ is undergoing the symmetry-breaking phase transition and slowly rolling towards φ = ± η . The model was based on scalar field dynamics obtained by re- placing the potential (1) by a symmetry breaking poten- tial of Coleman-Weinberg [16] form, where the mass term at the field origin is set to zero and symmetry breaking is obtained through quantum corrections. However, new inflation models typically suffer from an initial condition problem [17]. “Chaotic” (or “large-field”) inflation [18] is an alterna- tive scenario. Inflation is triggered by a period of slow- rolling of φ unrelated to a symmetry-breaking phase tran- sition. The simplest example occurs in the toy model of a single scalar field with potential where m is the mass of φ (which is of the order 10 − $^{6}$m$_{pl}$ if the model is to yield the observed magnitude of cos- mological fluctuations [2]). Here, m$_{pl}$ is the Planck mass defined via $_{m}$− 2 pl ≡ G , G being Newton’s gravitational constant. It is assumed that φ starts out at large field values and slowly rolls towards its vacuum state φ = 0. For inflation to be successful, two conditions must be sat- isfied. Firstly, the energy density must be dominated by the potential energy term, and secondly the acceleration term in the field equation the Klein-Gordon equation in an expanding background, must be negligible compared to the other two terms. Here, H is the Hubble expansion rate and a prime in- dicates the derivative with respect to φ . Making use of the Friedmann equation it is easy to see that the slow-rolling conditions are only satisfied for super-Planckian field values | φ | > m$_{pl}$ . In the above two equations (3) and (4) we have taken the field configuration to be homogeneous. In the case of chaotic inflation, the homogeneous slow-roll trajectory is V ( φ ) = 1 2 m $^{2}$φ2 (2) ¨ φ + 3 H ˙ φ = − V $^{′}$( φ ) , (3) H 2 = 8 πG 3 ( 1 2 ˙ φ 2 + V ( φ ) (4) V ( φ ) = 1 4 λ ( φ 2 − η $^{2}$)2 (1) II. INFLATION MODELS AND INITIAL CONDITIONS FOR REHEATING