This talk introduces the high-order proximal operator (HOPE) and high-order Moreau envelope (HOME) in the nonconvex setting. On the one hand, we establish the fundamental properties of HOPE and HOME, particularly the single-valuedness of HOPE and the differential properties of HOME under several conditions, e.g., prox-regularity or weak-convexity of the objective function. Moreover, the Hölder continuity of HOPE and gradient of HOME are discussed. On the other hand, this analysis serves as the basis for developing an inexact proximal point and an inexact gradient method in the nonconvex setting for which we study their convergence analysis. In particular, we discuss their linear convergence under Kurdyka-Łojasiewicz conditions of the original function. Some preliminary numerical results validate our theoretical foundations.