RESUMEN
Using backwards ray tracing, we study the shadows of Kerr black holes with scalar hair (KBHSH). KBHSH 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) noncompact, which yield not multiple images, (ii) compact, which produce an increasing number of Einstein rings and multiple images of the whole celestial sphere, and (iii) ultracompact, which possess light rings, yielding an infinite number of images with (we conjecture) a self-similar structure. The shadows of KBHSH, for Kerr-like horizons and noncompact BS-like hair, are analogous to, but distinguishable from, those of comparable Kerr BHs. But for non-Kerr-like horizons and ultracompact BS-like hair, the shadows of KBHSH are drastically different: novel shapes arise, sizes are considerably smaller, and multiple shadows of a single BH become possible. Thus, KBHSH provide quantitatively and qualitatively new templates for ongoing (and future) very large baseline interferometry observations of BH shadows, such as those of the Event Horizon Telescope.