The behavior of strongly interacting matter under extreme conditions is relevant for a variety of physical systems ranging from neutron stars to the early Universe. The physics of quarks and gluons in these settings can be investigated by means of first-principles lattice QCD simulations.  In this talk, I will give an introduction to some of the basic elements of lattice QCD. Moreover, I will present a selected set of results, focusing on the impact of high temperatures and nonzero isospin asymmetry between the light quark densities, and discuss their possible implications for neutron stars and cosmology.

(Szilárd Leó Colloquium)