Left behind after the supernova explosion of a massive star, a neutron star is the tiny, dense smoldering core of a once enormous object. These incredible stars are home to some of the most extreme conditions in the universe, with magnetic fields and gravity so high that our standard understandings do not apply.  Young neutron stars (called pulsars) are known to emit radiation across the entire electromagnetic spectrum, from radio waves to gamma rays.  I will discuss the observational history of these objects, ongoing projects to study them, what we have already learned, and what still remains a mystery after several decades of research.