Both parametric and nonparametric necessary and sufficient optimality conditions are established for a class of nonsmooth continuous-time programming problems. Subsequently, the structure and contents of these optimality results are utilized as a basis for constructing two parametric and four parameter-free duality models, and for proving weak, strong, and strict converse duality theorems. Furthermore, it is briefly pointed out how similar optimality and duality results can be derived for two special cases of the main problem containing arbitrary norms and square roots of positive semidefinite quadratic forms. These results generalize a number of similar results in the area of continuous-time programming, and subsume a great variety of cognate results originally developed for finite-dimensional nonlinear and fractional programming problems.