The measurement of solar oblateness has a rich history extending well back into the past. Until recently, its estimate has been actively disputed, as has its temporal dependence. Recent accurate observations of the solar shape gave cause for doubt, and so far only balloon flights or satellite experiments, such as those onboard SDO, seem to achieve the required sensitivity to measure the expected small deviations from sphericity. A shrinking or an expanding shape is ultimately linked to solar activity (likely not homologously with its change), as gravitational or magnetic fields, which are existing mechanisms for storing energy during a solar cycle, lead to distinct perturbations in the equilibrium solar-structure and changes in the diameter. It follows that a sensitive determination of the solar radius fluctuations might give information about the origin of the solar cycle. In periods of higher activity, the outer photospheric shape seems to become aspheric under the influence of higher-order multipole moments of the Sun, resulting both from the centrifugal force and the core rotation. An accurate determination of the shape of the Sun is thus one of the ways that we have now for peering into its interior, learning empirically about flows and motions there that would otherwise only be guessed at from theoretical considerations, developing more precise inferences, and ultimately building possible alternative gravitational theories.