  Motivated by the recent discovery of higher-order topological insulators, we
study their counterparts in strongly interacting bosons: `higher-order symmetry
protected topological (HOSPT) phases'. While the usual (1st-order) SPT phases
in d spatial dimensions support anomalous (d-1)-dimensional surface states,
HOSPT phases in d dimensions are characterized by topological boundary states
of dimension (d-2) or smaller, protected by certain global symmetries and
robust against disorders. Based on a dimensional reduction analysis, we show
that HOSPT phases can be built from lower-dimensional SPT phases in a way that
preserves the associated crystalline symmetries. When the total symmetry is a
direct product of global and crystalline symmetry groups, we are able to
classify the HOSPT phases using the K\"unneth formula of group cohomology.
Based on a decorated domain wall picture of the K\"unneth formula, we show how
to systematically construct the HOSPT phases, and demonstrate our construction
with many examples in two and three dimensions.
