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A general existence and uniqueness result of Picard-Lindelöf type is proved for ordinary differential equations in Fréchet spaces as an application of a generalized Nash-Moser implicit function theorem. Many examples show that the assumptions of the main result are natural. Applications are given for the Fréchet spaces $C^{\infty }\left(K\right)$, $S\left({ℝ}^N\right)$, $B\left(ℝR^N\right)$, $D_{L_1}\left({ℝ}^N\right)$, for Köthe sequence spaces, and for the general class of subbinomic Fréchet algebras.