Abstract: Axions and other similar particles can have unique cosmological histories that differ drastically from standard WIMP-like cold dark matter.  For example, they may condense into compact Bose-Einstein condensate states known as stars or miniclusters.  I discuss several such models, their cosmological consequences and astrophysical observables.  The QCD axion is the canonical example and may form axion stars with a mass around 10^11 M⊙.  I point out that a brand new astrophysical object, a Hydrogen Axion Star (HAS), may well be formed by ordinary baryonic matter becoming gravitationally bound to an axion star. I discuss the properties of the HAS and find that the hydrogen cloud has a high pressure and temperature in the center and is likely in the liquid metallic hydrogen state. High resolution ultraviolet, optical and infrared telescopes can discover HAS via black-body radiation.  I then discuss recent developments and constraints on Majorons of spontaneous lepton number breaking, which can have similar properties to axions while having a different natural parameter space.  For both axion and Majoron stars, I discuss some subtleties in recent constraints from gravitational lensing.