In modern science, including theoretical physics, as in the early classical mechanics, the unnatural reversible time of Newton, based on the medieval concept of geometric time by Nicholas Oresme, is still used. This "original sin" of natural sciences has unintended consequences and creates a set of paradoxes and methodological problems for science. The book explores two new models of essentially irreversible time - decelerating cosmological time and irreversible discrete time of a microcosm. It discusses recent astronomical observations that reveal evidence of the cosmological deceleration of the pace of time in the distant cosmos, in the solar system and on earth. The structure of the model of irreversible discrete time of a microcosm, as considered in the book, allows for the existence of both time and anti-time. In particular, the model predicts new uncertainty relations and violation of the mirror symmetry of the integral internal parity of the entire population of micro particles that correspond to current studies of elementary particle physics.