The "Lefschetz Principle" is the informal idea that for geometric questions about algebraic varieties over fields of characteristic 0, it is often sufficient to assume the ground field is the complex numbers (where analytic tools are available). The precise formulation of this principle makes it possible to use positive-characteristic geometry or number theory to solve problems concerning projective complex-analytic manifolds (and in some cases no solution by complex-analytic methods is known).  In analytic geometry over a non-archimedean field, it is often convenient (and sufficient) to consider algebraically closed ground fields, but the price to pay is that powerful arithmetic and algebraic techniques available over "smaller" non-archimedean fields cannot be used.  This led Scholze to ask (in the spirit of the Lefschetz Principle): can questions about "compact manifolds" over general non-archimedean fields be systematically reduced to the case of "smaller" fields where arithmetic techniques may be applied? 
After reviewing some links between complex algebraic geometry and number theory, I'll discuss the meaning of geometry over a non-archimedean field and Scholze's motivating application, and present the affirmative answer to his question.  Even over fields of characteristic 0, the solution rests on geometry in positive characteristic.  This is joint work with O. Gabber.