This paper introduces and discusses semismooth and semiconvex functions in the context of nonsmooth, nonconvex constrained optimization problems. The author defines these functions and their properties, including their generalized gradients. An optimization algorithm is presented that uses these gradients to converge to stationary points if the functions are semismooth. If the functions are also semiconvex and a constraint qualification is satisfied, the stationary point is shown to be optimal. The paper also explores the pointwise maximum of continuously differentiable and semiconvex functions, demonstrating that these operations preserve semismoothness and semiconvexity, respectively. Additionally, it shows that the composition of semismooth functions is semismooth and provides a chain rule for generalized gradients. The paper concludes with a discussion on stationarity and optimality conditions, proving that stationarity implies optimality under certain conditions.This paper introduces and discusses semismooth and semiconvex functions in the context of nonsmooth, nonconvex constrained optimization problems. The author defines these functions and their properties, including their generalized gradients. An optimization algorithm is presented that uses these gradients to converge to stationary points if the functions are semismooth. If the functions are also semiconvex and a constraint qualification is satisfied, the stationary point is shown to be optimal. The paper also explores the pointwise maximum of continuously differentiable and semiconvex functions, demonstrating that these operations preserve semismoothness and semiconvexity, respectively. Additionally, it shows that the composition of semismooth functions is semismooth and provides a chain rule for generalized gradients. The paper concludes with a discussion on stationarity and optimality conditions, proving that stationarity implies optimality under certain conditions.