Let C^(n) be the proper class of cardinals kappa that are Sigma_n-correct in V, meaning that V_kappa is a Sigma_n-elementary substructure of V. We will consider several types of large cardinal notions obtained by requiring that both the critical point kappa of an elementary embedding j:V -> M and its image under j are in C^(n). These are the C^(n)-cardinals. One of the results is that Vopenka's Principle is equivalent to the existence of a C^(n)-extendible cardinal, for every n.