Serre's result GAGA establishes a correspondence between algebraic and analytic sheaves, and it can be used to prove algebraic results by analytic methods (e.g. the realization of groups as Galois groups of algebraic branched covers of complex curves). Grothendieck's Existence Theorem plays an analogous role relating algebraic and formal sheaves, and can be used to study curves and their covers over fields other than the complex numbers. Not only are these two results analogous, but so are their proofs. In this talk and a sequel (in three weeks), these results and their parallels will be discussed.