  In this work, we introduce a new type of topological order which is protected
by subsystem symmetries which act on lower dimensional subsets of lattice
many-body system, e.g. along lines or planes in a three dimensional system. The
symmetry groups for such systems exhibit a macroscopic number of generators in
the infinite volume limit. We construct a set of exactly solvable models in
$2d$ and $3d$ which exhibit such subsystem SPT (SSPT) phases with one
dimensional subsystem symmetries. These phases exhibit analogs of phenomena
seen in SPTs protected by global symmetries: gapless edge modes, projective
realizations of the symmetries at the edge and non-local order parameters. Such
SSPT phases are proximate, in theory space, to previously studied phases that
break the subsystem symmetries and phases with fracton order which result upon
gauging them.
