Markov Random Flights is the first systematic presentation of the theory of Markov random flights in the Euclidean spaces of different dimensions. Markov random flight is a stochastic dynamic system subject to the control of an external Poisson process and represented by the stochastic motion of a particle that moves at constant finite speed and changes its direction at random Poisson time instants. The initial (and each new) direction are taken at random according to some probability distribution on the unit sphere. Such stochastic motion is the basic model for describing many real finite-velocity transport phenomena arising in statistical physics, chemistry, biology, environmental science, financial markets. Markov random flights acts as an effective tool for modeling the slow and super-slow diffusion processes arising in various fields of science and technology. Features Provides the first systematic presentation of the theory of Markov random flights in the Euclidean spaces of different dimensions Suitable for graduate students and for specialists and professionals in applied areas Introduces a new unified approach based on the powerful methods of mathematical analysis, such as integral transforms, generalized, hypergeometric and special functions.