I study the spectral properties of the almost Mathieu operator (AMO)—a well-known and well-studied self-adjoint bounded linear operator on l^2(Z) that is also a one-dimensional discrete Schrodinger operator. To begin with, by exploiting the chiral gauge transformation of the AMO, I derive a new formula in the style of the celebrated Chambers formula. This result, together with a perturbation theory approach based on Lidskii’s inequalities, is then used to obtain a sharper upper bound for the Lebesgue measure of the spectrum of the AMO in the case of rational frequency and critical coupling. I continue by establishing several estimates for the sizes of spectral gaps and bands, again for the case of rational frequency, but this time specifically for frequency having odd denominator. This latter work is conducted for a general (i.e., possibly non-critical) coupling constant, thereby generalising some key critical-case results of Krasovsky. The dissertation concludes with an appendix containing a rigorous proof for a variational principle due to Weyl that is occasionally quoted without reference in literature on the AMO.