  We study various non-relativistic field theories with exotic symmetries
called subsystem symmetries, which have recently attracted much attention in
the context of fractons. We start with a scalar theory called $\phi$-theory in
$d+1$ dimensions and discuss its properties studied in literature for $d\leq 3$
such as self-duality, vacuum structure, 't Hooft anomaly, anomaly inflow and
lattice regularization. Next we study a theory called chiral $\phi$-theory
which is an analogue of a chiral boson with subsystem symmetries. Then we
discuss theories including fermions with subsystem symmetries. We first
construct a supersymmetric version of the $\phi$-theory and dropping its
bosonic part leads us to a purely fermionic theory with subsystem symmetries
called $\psi$-theory. We argue that lattice regularization of the $\psi$-theory
generically suffers from an analogue of doubling problem as previously pointed
out in the $d=3$ case. We propose an analogue of Wilson fermion to avoid the
``doubling" problem. We also supersymmetrize the chiral $\phi$-theory and
dropping the bosonic part again gives us a purely fermionic theory. We finally
discuss vacuum structures of the theories with fermions and find that they are
infinitely degenerate because of spontaneous breaking of subsystem symmetries.
