In this review paper, I focus on presenting the reasons of extending the partial diferen‑
tial equations to space-time fractional diferential equations. I believe that extending
any partial diferential equations or any system of equations to fractional systems with‑
out giving concrete reasons has no sense. The experiments agrees with the theoretical
studies on extending the frst order-time derivative to time-fractional derivative. The
simulations of some processes also agrees with the theory of continuous time random
walks for extending the second-order space fractional derivative to the Riesz–Feller
fractional operators. For this aim, I give a condense review the theory of Brownian
motion, Langevin equations, difusion processes and the continuous time random
walk. Some partial diferential equations that are successfully extended to space-timefractional diferential equations are also presented.