The Mordell Conjecture (proved by Faltings in 1983) is a landmark result exemplifying the philosophy "Geometry controls arithmetic". However, it also implies that we can never hope to understand the arithmetic of a higher genus curve solely by studying its rational points over a fixed number field. In this talk, we will introduce the concepts of parametrized points, isolated points, and density degree sets and show how they, together with the Mordell-Lang conjecture (proved by Faltings in 1994), allow us to organize all algebraic points on a curve.