Quantum field theory in curved space-time and quantum gravity predict one-loop quantum corrections to the R+R^2 (Starobinsky) inflationary model. Observational consequences of a number of such corrections are considered. One of them – the form of the Fourier power spectrum of primordial scalar (matter density) perturbations – has been actually measured and has appeared to be in the excellent agreement with the theoretical prediction. The model also makes the definite prediction for the tensor-to-scalar ratio r=3(1-n_s)^2=0.004 which serves as a target for future observational search. The action of these corrections after inflation provides the decay of the effective scalar particle in this model (dubbed scalaron) into pairs of particles and anti-particles of known quantum fields (but not into pairs of gravitons), that is necessary for internal consistency of the model. Finally, effect of these corrections on the background dynamics during inflation can be calculated and compared with data. In this case no noticeable effect is found, and only upper bounds on quantum corrections are obtained. This has been expected since the relative smallness of these corrections is caused by the appearance of new hierarchy - the anomalously large value of the dimensionless coefficient in front of the R^2 term that finally follows from the actual amount of present large-scale inhomogeneity of the Universe. The large value of the effective R^2 term compared to the Weyl squared one also makes possible to consider the model non-perturbatively in the scalar sector (where it is renormalizable) for curvatures much less than the Planck one, while corrections in the tensor sector are still small.