I review the present status of the inflationary scenario in the early Universe and the main observational evidence on which it is based. The very possibility to make definite statements and predictions about a new historical stage in the past of our Universe which preceded the hot Big Bang is based on the existence of its artifacts in the present Universe – those Fourier modes of small scalar and tensor perturbations of the isotropic homogeneous Universe which amplitude is approximately constant in the super-Hubble regime independently of unknown physics at very high energies and curvatures (still assuming that the local Lorentz invariance is not broken). The existence of such modes was first proved by Lifshitz in the perturbation theory long time ago and then generalized to arbitrary amplitudes of metric perturbations in the scope of the Lifshitz-Khalatnikov quasi-isotropic solution (1960). The primordial Fourier spectrum of such scalar perturbations at cosmological scales has been measured by now using temperature anisotropy and polarization of the cosmic microwave background and other methods.
Confirming the generic prediction of slow-roll inflationary models, this spectrum appears to be approximately scale-invariant, but its deviation from scale invariance has been reliably measured, too. The numerical value of this deviation (n_s-1)\approx -0.035 leaves viable only a few of slow-roll inflationary models. The simplest of them based either on scalar fields in General Relativity or on modified f(R) gravity, which produce the best fit to all existing astronomical data, require one, maximum two dimensionless parameters taken from observations only. The main discoveries expected for these models in future are discussed. Among them the most fundamental are primordial quantum gravitational waves generated during inflation. In one parametric models, including the original R+R^2 one (1980), the definite prediction for the tensor-to-scalar ratio r=3(1-n_s)^2=0.004 follows. In the models considered, the most generic predecessor of inflation is an anisotropic and inhomogeneous space-time near a generic space-like singularity similar to that first studied by Lifshitz and Khalatnikov. Since the transition from such space-time to the generalized quasi-de Sitter regime is generic, too, for inflation to begin inside a patch including the observable part of the Universe, causal connection inside the whole patch is not necessary. However, it becomes obligatory for a graceful exit from inflation in order to have practically the same number of e-folds during inflation inside this patch.