In this work, starting from foreknowledge of probability theory, we have dealt with probabilities of rare events, respectively with asymptotic estimates of rare events. We have given an elementary introduction to large deviations, highlighting the role of Crámer and Varadhan. As introduction to the main subject we have presented some elementary results from probability theory and metric spaces. We have proved Crámer’s theorem, which provides a general principle to compute the rate function I. We have shown that the rate function I is the Legendre-Fenchel transform of the function \(\ln\varphi\). We have developed a formal framework for the quantification of probabilities of rare events in which the complete theory of large deviations can be developed. We have shown that Crámer’s theorem is essentially large deviation principle. We have formulated important general theorem for large deviation theory, generalization of the Laplace method, Varadhan’s lemma. In the end we have shown that very weak version of Stirling’s formula follows from Varadhan’s lemma.