  In the new era of gravitational wave (GW) and multi-messenger astrophysics,
the detection of a GW signal from the coalescence of a black hole - neutron
star (BHNS) binary remains a highly anticipated discovery. This kind of system
is expected to be within reach of the second generation of ground-based
detectors. In this context, we develop a series of versatile semi-analytical
models to predict the properties of all the electromagnetic (EM) counterparts
of BHNS mergers. We include the nuclear-decay-powered kilonova emission, its
radio remnant, the prompt emission from the jet and the related afterglow. The
properties of these counterparts depend upon those of the outflows that result
from the partial disruption of the NS during the merger and from the accretion
disc around the remnant, which are necessary ingredients for transient EM
emission to accompany the GW signal. We therefore define ways to relate the
properties of these outflows to those of the progenitor binary, establishing a
link between the binary parameters and the counterpart properties. From the
resulting model, we anticipate the variety of light curves that can emerge
after a BHNS coalescence, from the radio up to gamma-rays. These light curves
feature universal traits which are the imprint of the dynamics of the emitting
outflows, but at the same time they show a clear dependence on the BH mass and
spin, though with a high degree of degeneracy. The latter can be deduced by
joint GW - EM analysis. In this paper, we perform a proof-of-concept
multi-messenger parameter estimation of a BHNS merger with an associated
kilonova, to test how the information from the EM counterpart can complement
that from the GW signal. Our results indicate that the observation and modeling
of the kilonova can help to break the degeneracies in the GW parameter space,
leading to better constraints on, e.g., the BH spin.
