Axions (and other light bosons) with masses in the range $10^{-33}\text{ eV}\lesssim m\lesssim 10^{-18}\text{ eV}$ have distinctive effects on cosmology. These effects allow one to observationally distinguish ultralight axions from the other major cosmological components of dark energy, cold dark matter, warm dark matter, and neutrinos. Current observations of the cosmic microwave background and large scale structure place percent-level constraints on the axion energy density, and limit them to be sub-dominant to cold dark matter for $m\lesssim 10^{-24}\text{ eV}$. Future observations with CMB-S4 will improve these constraints by up to two orders of magnitude and allow one to distinguish the effects of axions from those of massive standard model neutrinos. This allows sub-percent level tests of the one-species cold dark matter paradigm over a vast range of cosmic scales. Furthermore, axions with $m\approx 10^{-22}\text{ eV}$ could play a role in resolving issues of small-scale structure formation with cold dark matter, and are also distinct from warm dark matter. Efforts to probe this mass range (and higher) observationally include using high redshift galaxy formation, cosmic reionization, the Lyman-alpha forest flux power spectrum, CMB lensing, galaxy weak lensing, the 21cm power spectrum, and galactic substructure. In the lab, such axions and other bosons induce effects with long time scale oscillations, which could be probed with upcoming experiments such as CASPEr.