Birational geometry of del Pezzo surfaces

A del Pezzo surface is a smooth projective surface with ample anticanonical divisor. Over an algebraically closed field, any surface like this is rational. However, without this assumption del Pezzo surfaces exhibit very interesting birational properties. I will survey some old and new results about birational geometry of del Pezzo surfaces over arbitrary fields, mostly focusing on Severi–Brauer surfaces, quadrics, and del Pezzo surfaces of degree 4.