The paper discusses the Mittag-Leffler functions and their applications in various scientific and engineering fields. These functions, introduced by the Swedish mathematician G. M. Mittag-Leffler in 1903, have gained significant importance in the last two decades due to their wide range of applications in solving problems related to fractional calculus, physics, biology, and engineering. The paper presents a detailed account of the Mittag-Leffler function, its generalizations, and related functions, along with their properties and applications. It also provides a comprehensive survey of the literature on these functions, highlighting their role in fractional differential and integral equations, as well as in the study of complex systems. The paper is structured into several sections, starting with an introduction to the Mittag-Leffler function and its special cases. It then discusses functional relations, basic properties, recurrence relations, asymptotic expansions, and integral representations of the Mittag-Leffler functions. The paper also explores the connection between the Mittag-Leffler functions and the Riemann-Liouville fractional calculus operators, as well as their applications in solving fractional differential equations and other problems in physics and engineering. The paper concludes with a discussion of the generalized Mittag-Leffler functions and their properties, as well as their applications in various fields. The authors emphasize the importance of the Mittag-Leffler functions in the study of fractional-order systems and their potential for further research and development.The paper discusses the Mittag-Leffler functions and their applications in various scientific and engineering fields. These functions, introduced by the Swedish mathematician G. M. Mittag-Leffler in 1903, have gained significant importance in the last two decades due to their wide range of applications in solving problems related to fractional calculus, physics, biology, and engineering. The paper presents a detailed account of the Mittag-Leffler function, its generalizations, and related functions, along with their properties and applications. It also provides a comprehensive survey of the literature on these functions, highlighting their role in fractional differential and integral equations, as well as in the study of complex systems. The paper is structured into several sections, starting with an introduction to the Mittag-Leffler function and its special cases. It then discusses functional relations, basic properties, recurrence relations, asymptotic expansions, and integral representations of the Mittag-Leffler functions. The paper also explores the connection between the Mittag-Leffler functions and the Riemann-Liouville fractional calculus operators, as well as their applications in solving fractional differential equations and other problems in physics and engineering. The paper concludes with a discussion of the generalized Mittag-Leffler functions and their properties, as well as their applications in various fields. The authors emphasize the importance of the Mittag-Leffler functions in the study of fractional-order systems and their potential for further research and development.