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Galaxies are three-dimensional (3D) structures moving under the dictates of gravity in a 3D Universe. From our position on the Earth, astronomers have only the opportunity to observe their properties projected onto a two-dimensional (2D) plane, usually called the plane of the sky. Since we can neither circumnavigate galaxies nor wait until they spin around, our knowledge of the intrinsic shape of galaxies is still limited, relying on sensible, but sometimes not accurate, physical and geometrical hypotheses.

Despite the obvious difficulties inherent to measure the intrinsic 3D shape of galaxies, it is doubtless that it keeps an invaluable piece of information about their formation and evolution. In fact, astronomers have acknowledged this since galaxies were established to be island universes and the topic has produced an outstanding amount of literature during the last century.

In this paper I discuss the main developments and results in the quest to better understand the 3D shape of galaxy bulges. Given the limited space available in this chapter, I have not elaborated on the concept and definition of a bulge, leaving this discussion to another chapter in this volume. In the same way, I have deliberately not included the intrinsic shape of boxy/peanut (B/P) structures located in the centre of disc galaxies which some authors associate to galaxy bulges (Lütticke, Dettmar & Pohlen, 2000). Currently it is well established that these structures are actually part of the bar and intimately related to their secular evolution (Combes & Sanders, 1981, Chung & Bureau, 2004). As bars evolve, stars can be moved perpendicular to the disc plane due to a coherent bending of the bar producing its characteristic shape (Debattista et al., 2004, Martinez-Valpuesta, Shlosman & Heller, 2006). B/P structures share the same photometric and kinematic properties of bars (Méndez-Abreu et al., 2008b, Erwin & Debattista, 2013).

On the other hand, I have included a historical review of the evolution of our knowledge of the intrinsic shape of elliptical galaxies. The properties of elliptical galaxies and those of intermediate/massive galaxy bulges have been often considered to be similar (Wyse, Gilmore & Franx, 1997). This is particularly true when referring to their surface-brightness distributions and shapes. Indeed, it has been common in the literature to rely on both simulations and observations of elliptical galaxies to interpret the observational properties of bulges (e.g., Kormendy & Bender, 2012).

This paper is structured as follows. In Section 2 I describe the basic geometric considerations of the problem and set up the notation used throughout the chapter. In Section 3 I review our current knowledge on the intrinsic shape of both elliptical and disc galaxies. Section 4 introduces the advantages and drawbacks of studying galaxy bulges with respect to ellipticals and a historical perspective of their 3D shape measurements. In Section 5 I summarize the evolution of the concept of the Milky Way bulge and its intrinsic 3D shape. Section 6 addresses the importance of numerical simulations to understand the physical processes that shape galaxy ellipsoids. Finally, in Section 7 I sketch out the current view on the intrinsic shape of bulges and explore future prospects.

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