  Close-in exoplanets interact with their host stars gravitationally as well as
via their magnetized plasma outflows. The rich dynamics that arises may result
in distinct observable features. Our objective is to study and classify the
morphology of the different types of interaction that can take place between a
giant close-in planet (a Hot Jupiter) and its host star, based on the physical
parameters that characterize the system. We perform 3D magnetohydrodynamic
numerical simulations to model the star--planet interaction, incorporating a
star, a Hot Jupiter, and realistic stellar and planetary outflows. We explore a
wide range of parameters and analyze the flow structures and magnetic
topologies that develop. Our study suggests the classification of star--planet
interactions into four general types, based on the relative magnitudes of three
characteristic length scales that quantify the effects of the planetary
magnetic field, the planetary outflow, and the stellar gravitational field in
the interaction region. We describe the dynamics of these interactions and the
flow structures that they give rise to, which include bow shocks, cometary-type
tails, and inspiraling accretion streams. We point out the distinguishing
features of each of the classified cases and discuss some of their
observationally relevant properties. The magnetized interactions of
star--planet systems can be categorized, and their general morphologies
predicted, based on a set of basic stellar, planetary, and orbital parameters.
