Many important questions in Diophantine geometry seek to understand the algebraic properties of important transcendental functions, such as the exponential function. Among these questions are the Existential Closedness problems, which ask when solutions exist to a system of polynomial equations that involve said transcendental functions. In this talk, I will introduce these problems and the various functions we are interested in, explain the partial results that have been proven so far, and show how they fit in with other important theorems and conjectures.