This volume is based on a course of lectures delivered at the St. Petersburg State University (Russia) and at Ohio State University (Columbus). It is intended as a textbook for graduate students and postdocs. The book presents the theory of records and some information on order statistics. Also included are exercises illustrating the examples and developing the ideas. The past 20 years has seen tremendous progress in the topic, giving forth a large number of new models that reflect the dynamics of records in a wide range of areas. This volume presents systematic main results with a special emphasis on non-classical record schemes. The material is presented in a comprehensive style succinctly outlining the current state of the theory. The work is geared toward statisticians, actuarians, engineers, hydrologists, meteorologists, and sports and market analysts.