In this paper, we review some of the main results in the field of abstract nonlinear fractional evolution equations. We study solutions of the semilinear Cauchy problem in the subdiffusive and superdiffusive cases, mainly with the Caputo and Riemann-Liouville fractional-order derivative, in the setting of the real semiaxis and real axis, and under various assumptions on the main data of the given equations. We consider in our analysis several kinds of perturbed systems, for example, delay, control, and stochastic properties, even with nonlocal conditions and impulses. We provide a complete description of the representation of mild solutions in terms of associated solution families of operators.
