Entanglement in curved spacetimes and cosmology

We review recent results regarding entanglement in quantum fields in cosmological spacetimes and related phenomena in flat spacetime such as the Unruh effect. We being with a summary of important results about field entanglement and the mathematics of Bogoliubov transformations that is very often used to describe it. We then discuss the Unruh-DeWitt detector model, which is a useful model of a generic local particle detector. This detector model has been successfully used as a tool to obtain many important results. In this context we discuss two specific types of these detectors: a qubit and a harmonic oscillator. The latter has recently been shown to have important applications when one wants to probe nonperturbative physics of detectors interacting with quantum fields. We then detail several recent advances in the study and application of these ideas, including echoes of the early universe, entanglement harvesting, and a nascent proposal for quantum seismology.