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The spectrum of the Hill operator (one-dimensional periodic Schr\"{o}dinger operator) $L=-d^2/dx^2+v$ where $v\in L^2([0,\pi])$, has a band structure, that is, the intervals of continuous spectrum alternate with \textit{spectral gaps}, or \textit{instability zones}. The sizes of these zones decay, and the rate of decay depends on the smoothness of the potential. In the opposite direction, one can make conclusions about the smoothness of a potential based on the rate of decay of the instability zones. In this talk, together with main results in the literature, a Fourier method for studying the spectral properties of Hill operators will be presented