We present several integrable systems associated with nonlocal versions of the Hirota equations, which are important in the modeling of physical phenomena because they include both the nonlinear Schr\odinger (NLS) equation and the complex modified Korteweg--de Vries (cmKdV) equation. The integrability of these models is established through the explicit construction of their Lax pairs, or equivalently, their zero-curvature representations. The nonlocal Hirota equations are derived within the AKNS scheme via Ablowitz--Musslimani type nonlocal reductions. In addition, by constructing the Darboux transformation for the systems under consideration, we obtain exact solutions in explicit form.