  We present an effective field theory describing the relevant interactions of
the Standard Model with an electrically neutral particle that can account for
the dark matter in the Universe. The possible mediators of these interactions
are assumed to be heavy. The dark matter candidates that we consider have spin
0, 1/2 or 1, belong to an electroweak multiplet with arbitrary isospin and
hypercharge and their stability at cosmological scales is guaranteed by
imposing a $\mathbb{Z}_2$ symmetry. We present the most general framework for
describing the interaction of the dark matter with standard particles, and
construct a general non-redundant basis of the gauge-invariant operators up to
dimension six. The basis includes multiplets with non-vanishing hypercharge,
which can also be viable DM candidates. We give two examples illustrating the
phenomenological use of such a general effective framework. First, we consider
the case of a scalar singlet, provide convenient semi-analytical expressions
for the relevant dark matter observables, use present experimental data to set
constraints on the Wilson coefficients of the operators, and show how the
interplay of different operators can open new allowed windows in the parameter
space of the model. Then we study the case of a lepton isodoublet, which
involves co-annihilation processes, and we discuss the impact of the operators
on the particle mass splitting and direct detection cross sections. These
examples highlight the importance of the contribution of the various
non-renormalizable operators, which can even dominate over the gauge
interactions in certain cases.
