I will discuss some of the driving questions in the area of Rational points on algebraic varieties (these can be thought of as solutions over fields of numbertheoretic interest to systems of homogeneous polynomial equations). I will focus on the case of smooth rational surfaces, and discuss some results concerning the arithmetic of del Pezzo surfaces of degree 1. I will also explain how these results complete a qualitative picture of basic arithmetic phenomena among smooth rational surfaces. Along the way I will go over concepts like weak approximation and the computation of Brauer-Manin obstructions; I will not assume previous knowledge of them.