In this talk we will consider the spectrum of the Almost Mathieu operator, \[ \left(H_{b,\phi} x\right)_n= x_{n+1} +x_{n-1} + b \cos\left(2 \pi n\omega + \phi\right)x_n, \] on $l^2(\mathbb{Z})$. We will show that for $b \ne 0,\pm 2$ and $\omega$ Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called ``Ten Martini Problem'', for these values of $b$ and $\omega$.

The proof uses a combination of results on reducibility, localization and duality for the Almost Mathieu operator, and its associated eigenvalue equation, sometimes called the Harper equation.

Finally, we will also show that for $|b|\ne 0$ small enough or large enough all spectral gaps predicted by the Gap Labelling theorem are open.