Abstract: Brown representability tells us there’s a bijection between generalized cohomology theories and spectra, and that morphisms of cohomology theories come from morphisms of the corresponding spectra. So is the category of cohomology theories equivalent to the stable homotopy category? No! In the spirit of Halloween there are  phantom  maps  that are both dead and alive ️— nonzero maps of spectra inducing the zero map on cohomology. In this spooky talk I’ll discuss phantom maps and related topics including their non-stable analogues - non-nullhomotopic maps of CW complexes which kill each finite skeleton. Costumes encouraged.