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Structural relaxation in deeply supercooled liquids is non-exponential. In susceptibility representation, $\chi^{\prime\prime}(\nu)$, the spectral shape of the structural relaxation is observed as an asymmetrically broadened peak with a $\nu^{1}$ low- and $\nu^{-\beta}$ high-frequency behavior. In this perspective article we discuss common notions, recent results and open questions regarding the spectral shape of the structural relaxation. In particular, we focus on the observation that a high-frequency behavior of $\nu^{-1/2}$ appears to be a generic feature in a broad range of different deeply supercooled liquids. Moreover, we review extensive evidence that contributions from orientational cross-correlations can lead to deviations from the generic spectral shape in certain substances, in particular in dielectric loss spectra. Additionally, intramolecular dynamics can contribute significantly to the spectral shape in substances containing more complex and flexible molecules. Finally, we discuss the open questions regarding potential physical origins of the generic $\nu^{-1/2}$ behavior and the evolution of the spectral shape towards higher temperatures.