We develop algebraic methods for study of homogeneous submanifolds in homogeneous spaces. This includes the description of stationary subgroups of $k$-jets of homogeneous submanifolds, algebraic version of Cartan's method of the moving frame, description of subalgebras corresponding to symmetry groups of homogeneous submanifolds. We also develop several algebraic techniques for classification of homogeneous submanifolds. As application we present the results of such classifications in low-dimensional affine and projective geometries.