We review the fundamental theory of solution operators associated to abstract linear fractional evolution equations. We provide their basic results concerning generation, analyticity, and inversion. We show the subordination principle as well as perturbation, approximation, ergodicity, and compactness properties. The immediate norm continuity is also analyzed. We show how this functional-operator machinery can be applied to solve the fractional-order Cauchy problem. We finish this review with some comments and open problems.
