We study atypically large fluctuations in non-equilibrium stochastic systems. Rare large fluctuations are important in many situations: from a sudden extinction of a long-lived population of animals to a rare large peak in the height of a growing surface, and to the unusual trajectory of the first among myriads of sperm cells, competing with each other for reaching the oocyte. The research involves a combination of analytical methods of theoretical physics and applied mathematics and numerical simulations.