Given two polynomials in two variables with no common factor, how many common zeros do they have? The answer, provided by the beautiful Bernstein-Kushnirenko theorem, involves the Newton polyhedra of the two polynomials, and an interesting notion from convex geometry called the mixed volume. In this talk I will begin by describing the Bernstein-Kushnirenko theorem, and then go on to introduce the recent works of Kaveh-Khovanskii and Lazarsfeld-Mustata, which uncovered further links between intersection numbers and (mixed) volumes, and explain how this interplay enriched our understanding for both concepts.