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We study the large scale geometry of the relative free splitting complex andthe relative complex of free factor systems of the rank $n$ free group $F_n$,relative to the choice of a free factor system of $F_n$, proving that thesecomplexes are hyperbolic. Furthermore we present the proof in a generalcontext, obtaining hyperbolicity of the relative free splitting complex and ofthe relative complex of free factor systems of a general group $\Gamma$,relative to the choice of a free factor system of $\Gamma$. The proof yieldsinformation about coarsely transitive families of quasigeodesics in each ofthese complexes, expressed in terms of fold paths of free splittings.

Relative free splitting and free factor complexes I: Hyperbolicity