  Using backwards ray tracing, we study the shadows of Kerr black holes with
scalar hair (KBHsSH). KBHsSH interpolate continuously between Kerr BHs and
boson stars (BSs), so we start by investigating the lensing of light due to
BSs. Moving from the weak to the strong gravity region, BSs - which by
themselves have no shadows - are classified, according to the lensing produced,
as: $(i)$ non-compact, which yield no multiple images; $(ii)$ compact, which
produce an increasing number of Einstein rings and multiple images of the whole
celestial sphere; $(iii)$ ultra-compact, which possess light rings, yielding an
infinite number of images with (we conjecture) a self-similar structure. The
shadows of KBHsSH, for Kerr-like horizons and non-compact BS-like hair, are
analogous to, but distinguishable from, those of comparable Kerr BHs. But for
non-Kerr-like horizons and ultra-compact BS-like hair, the shadows of KBHsSH
are drastically different: novel shapes arise, sizes are considerably smaller
and multiple shadows of a single BH become possible. Thus, KBHsSH provide
quantitatively and qualitatively new templates for ongoing (and future) very
large baseline interferometry (VLBI) observations of BH shadows, such as those
of the Event Horizon Telescope.
