Let x be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Following H. Wielandt, we call a subgroup H of a finite group G a submaximal x-subgroup if there exists an isomorphic embedding ϕ : G ↪ G * of G into some finite group G * under which G ϕ is subnormal in G * and H ϕ = K ∩ G ϕ for some maximal x-subgroup K of G *. In the case where x coincides with the class of all π-groups for some set π of prime numbers, submaximal x-subgroups are called submaximal π-subgroups. In his talk at the well-known conference on finite groups in Santa Cruz in 1979, Wielandt emphasized the importance of studying submaximal π-subgroups, listed (without proof) certain of their properties, and formulated a number of open questions regarding these subgroups. Here we prove properties of maximal and submaximal x- and π-subgroups and discuss some open questions both Wielandt’s and new ones. One of such questions due to Wielandt reads as follows: Is it always the case that all submaximal x-subgroups are conjugate in a finite group G in which all maximal x-subgroups are conjugate?