The quantum chromodynamics axion and axion-like particles are some of the most sought-after beyond the Standard Model particles at present because of their possible connections with the strong-CP problem, dark matter, and ultraviolet physics such as Grand Unification and String Theory.  Laboratory searches are underway around the world to search for these hypothetical particles.  However, these searches are notoriously difficult because the axion mass is currently unknown.  If the axion is generated after inflation, then the axion mass is in principle calculable by solving the axion dynamics in the early Universe, but such calculations are made difficult by nonlinear effects in the equations of motion.  I will present up-to-date calculations of the axion mass using adaptive mesh refinement simulations of axion dark matter in the early Universe.  I will then show that radio observations of neutron stars may be able to probe axion dark matter in the relevant mass range.  The early Universe simulations and astrophysical searches are part of a broader program, which I will overview, to rule out or confirm the existence of the axion in nature in the coming years.