The general theory of submanifolds in a multidimensional projective space is constructed in this book. The topics dealt with include osculating spaces and fundamental forms of different orders, asymptotic and conjugate lines, submanifolds on the Grassmannians, different aspects of the normalization problems for submanifolds (with special emphasis given to a connection in the normal bundle) and the problem of algebraizability for different kinds of submanifolds, and the geometry of hypersurfaces and hyperbands. A series of special types of submanifolds with special projective structures are studied: submanifolds carrying a net of conjugate lines (in particular, conjugate systems), tangentially degenerate submanifolds, and submanifolds with asymptotic and conjugate distributions. The method of moving frames and the apparatus of exterior differential forms are systematically used in the book and the results presented can be applied to the problems dealing with the linear subspaces or their generalizations. Graduate students majoring in differential geometry should find this monograph of interest, as should researchers in differential and algebraic geometry, complex analysis and theory of several complex variables.