For the Schrödinger operator $-\Delta + V$ on $L^2 (\mathbb{R}^n)$, let $N(V)$ be the number of bound states. We will review a few classical bounds for $N(V)$: Birman-Schwinger, Cwikel-Lieb-Rosenbljum, Birman-Solomjak. We will then present new bounds for $N(V)$ in dimension two. This work was motivated by a conjecture of Khuri, Martin and Wu.