  In this review, we present a self-contained introduction to axion-like
particles (ALPs) with a particular focus on their effects on photon
polarization: both theoretical and phenomenological aspects are discussed. We
derive the photon survival probability in the presence of photon--ALP
interaction, the corresponding final photon degree of linear polarization, and
the polarization angle in a wide energy interval. The presented results can be
tested by current and planned missions such as IXPE (already operative), eXTP,
XL-Calibur, NGXP, XPP in the X-ray band and like COSI (approved to launch),
e-ASTROGAM, and AMEGO in the high-energy range. Specifically, we describe
ALP-induced polarization effects on several astrophysical sources, such as
galaxy clusters, blazars, and gamma-ray bursts, and we discuss their real
detectability. In particular, galaxy clusters appear as very good observational
targets in this respect. Moreover, in the very-high-energy (VHE) band, we
discuss a peculiar ALP signature in photon polarization, in principle capable
of proving the ALP existence. Unfortunately, present technologies cannot detect
photon polarization up to such high energies, but the observational capability
of the latter ALP signature in the VHE band could represent an interesting
challenge for the future. As a matter of fact, the aim of this review is to
show new ways to make progress in the physics of ALPs, thanks to their effects
on photon polarization, a topic that has aroused less interest in the past, but
which is now timely with the advent of many new polarimetric missions.
