Cox rings generalize the homogeneous coordinate rings of projective spaces to varieties with finitely generated divisor class groups. We will introduce the basic definitions and properties in this theory. In the second part of the talk, we will study the Cox rings of each of the varieties in two families whose elements are toric surfaces blown up at a point. We will classify each variety in these two families according to whether its Cox ring is finitely generated. This talk is based on joint work with Javier Gonzalez and Kalle Karu.