  We study 't Hooft anomalies and the related anomaly inflow for subsystem
global symmetries. These symmetries and anomalies arise in a number of exotic
systems, including models with fracton order such as the X-cube model. As is
the case for ordinary global symmetries, anomalies for subsystem symmetries can
be canceled by anomaly inflow from a bulk theory in one higher dimension; the
corresponding bulk is therefore a non-trivial subsystem symmetry protected
topological (SSPT) phase. We demonstrate these phenomena in several examples
with continuous and discrete subsystem global symmetries, as well as
time-reversal symmetry. For each example we describe the boundary anomaly, and
present classical continuum actions for the corresponding bulk SSPT phases,
which describe the response of background gauge fields associated with the
subsystem symmetries. Interestingly, we show that the anomaly does not uniquely
specify the bulk SSPT phase. In general, the latter may also depend on how the
symmetry and the associated foliation structure on the boundary are extended
into the bulk.
