  The Higgs discovery and the lack of any other hint for new physics favor a
description of non-standard Higgs physics in terms of an effective field
theory. We present an implementation of a general Higgs effective Lagrangian
containing operators up to dimension six in the framework of FeynRules and
provide details on the translation between the mass and interaction bases, in
particular for three- and four-point interaction vertices involving Higgs and
gauge bosons. We illustrate the strengths of this implementation by using the
UFO interface of FeynRules capable to generate model files that can be
understood by the MadGraph 5 event generator and that have the specificity to
contain all interaction vertices, without any restriction on the number of
external legs or on the complexity of the Lorentz structures. We then
investigate several new physics effects in total rates and differential
distributions for different Higgs production modes, including gluon fusion,
associated production with a gauge boson and di-Higgs production. We finally
study contact interactions of gauge and Higgs bosons to fermions.
