We first present an overview on random motions at finite velocuty on the line an in the space. We study different forms of random motions, the probability distributions of which are solutions of the famous Euler-Poisson-Darboux equations which emerges also in the accelerated telegraph process. Finally, we present some fractional extensions of the finite-velocity random motions. References: 1) Fractional Klein-Gordon equations and related stochastic processes (2014) (with R. Garra and F. Polito) Journal of Statistical Physics, Vol. 155, 777-809. 2) Random flights governed by Klein-Gordon.type partial differential equations (2014) (with R. Garra). Stochastic Processes and their Appllications, Vol. 124, 2171-2187. 3) Flying randomly in R^d with Dirichlet displacements (2012) (with A. De Gregorio). Stochastic Processes and their Appllications, Vol. 122, 676-713. 4) Random flights in higher spaces (with A. De Gregorio) (2007). Journal of Theoretical Probability, Vol. 20, 769-806.