Summary: Both parametric and parameter-free saddle-point- and stationary-point-type necessary and sufficient optimality conditions are established for a class of continuous-time generalized fractional programming problems with convex operator inequality and affine operator equality constraints. Based on the forms and contents of these optimality results, a parameter-free Lagrangian-type, two parametric, and four parameter-free Wolfe-type duality models are constructed and appropriate duality theorems are proved. Furthermore, some of these optimality and duality results are specialized and briefly discussed for an important particular case of the main problem investigated in this paper.