These lecture notes aim to provide an introduction to entanglement entropies in quantum field theories, including holographic ones. We explore basic properties and simple examples of entanglement entropies, mostly in two dimensions, with an emphasis on physical rather than formal aspects of the subject. In the holographic case, the focus is on how that physics is realized geometrically by the Ryu-Takayanagi formula. In order to make the notes somewhat self-contained for readers whose background is in high-energy theory, a brief introduction to the relevant aspects of quantum information theory is included.