There are investigated elliptic pseudodifferential operators $p(x,D)$ which are degenerate on a submanifold $\Gamma$ of any codimension. Under certain further assumptions, for the operator which is obtained by adjoining to $p(x,D)$ boundary and coboundary conditions on the submanifold $\Gamma$ , there are constructed left and right regularizers, and theorems on hypoellipticity and local solvability are proved. In case $p(x,D)$ is defined on a smooth compact manifold it is shown to be noetherian on special weighted spaces of Sobolev type. Bibliography: 24 titles.