The existence and uniqueness of solutions to integral equations with a special degenerate kernel associated with the Caputo fractional integral is established using an approach based on fixed point theorems. Several properties of solutions are also examined, such as Hölder continuity, continuous dependence on the input data, and continuous dependence on the fractional order of the derivative. In addition, the invariance under time shifts in autonomous systems for solutions of fractional-order initial value problems, as well as explicit solution representations in terms of Mittag-Leffler functions (via the variation of constants formula), are considered. Finally, some results on the existence of solutions to boundary value problems for scalar Caputo fractional differential equations are presented.