Testing the equation of state of neutron stars with electromagnetic observations

Neutron stars are the densest, directly observable stellar objects in the universe and serve as unique astrophysical laboratories to study the behavior of matter under extreme physical conditions. This book chapter is devoted to describing how electromagnetic observations, particularly at X-ray, optical and radio wavelengths, can be used to measure the mass and radius of neutron stars and how this leads to constraints on the equation of state of ultra-dense matter. Having accurate theoretical models to describe the astrophysical data is essential in this effort. We will review different methods to constrain neutron star masses and radii, discuss the main observational results and theoretical developments achieved over the past decade, and provide an outlook of how further progress can be made with new and upcoming ground-based and space-based observatories.