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Since Mumford's 1979 construction proved the existence of fake projective planes, a natural problem has been to make these surfaces explicit: how can one actually write down defining equations for a fake projective plane? The history of this question passes through several major stages. First came Mumford's original existential construction via p-adic uniformization. This was followed by the arithmetic classification program: results of Klingler and Yeung showed that fake projective planes are arithmetic ball quotients, while subsequent work of Prasad-Yeung and Cartwright-Steger reduced the possibilities to a complete list of 50 cases up to isomorphism of fundamental groups, or 100 up to bi-holomorphism. A new phase began when Borisov and Keum produced the first explicit equations for a fake projective plane, opening the door to concrete geometric study of these surfaces. Since then, further explicit constructions have been obtained by several authors. In this talk I will survey this development, explain the main ideas behind the passage from existence to equations, and discuss possible interactions with fake quadrics, which may be viewed as a degree-two analogue in the broader landscape of ``fake'' surfaces. A new phase began when Borisov and Keum produced the first explicit equations for a fake projective plane, opening the door to concrete geometric study of these surfaces. Since then, further explicit constructions have been obtained by several authors. In this talk I will survey this development, explain the main ideas behind the passage from existence to equations, and discuss possible interactions with fake quadrics, which may be viewed as a degree-two analogue in the broader landscape of ``fake'' surfaces.