In this mini review, we present assumptions and some results of Monte Carlo simulations based on the Penna model, which support the mutation accumulation theory of aging. This microscopic model has been exploited for the quantitative description of many biological phenomena connected with the population evolution. We show how the results of simulations could describe the changes of mortality trajectories of the human populations during the last 150 years, and how the method could be used for predicting the human age distribution in the future. The main assumption of the model is that genes are expressed chronologically one after another in the same order in all individuals during their life span. Different forces of selection pressure exerted on genes expressed at different periods of life generate characteristic gradient of defective genes accumulated in the germline cells. The genes expressed after the minimum reproduction age are under weaker selection pressure, and the fraction of defects among them is higher than among the genes expressed before the minimum reproduction age. This gradient of defects generates a gradient of mortality for the part of the population in the reproduction age following the exponential Gompertz law. The limitations of the model and some biological interpretations of its parameters are also discussed.