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We introduce the basic elements of the theory of parametrized$\infty$-categories and functors between them. These notions are defined assuitable fibrations of $\infty$-categories and functors between them. We giveas many examples as we are able at this stage. Simple operations, such as theformation of opposites and the formation of functor $\infty$-categories, becomeslightly more involved in the parametrized setting, but we explain preciselyhow to perform these constructions. All of these constructions can be performedexplicitly, without resorting to such acts of desperation as straightening. Thekey results of this Expos\'e are: (1) a universal characterization of the$T$-$\infty$-category of $T$-objects in any $\infty$-category, (2) theexistence of an internal Hom for $T$-$\infty$-categories, and (3) aparametrized Yoneda lemma.

Parametrized higher category theory and higher algebra: Expos\'e I -- Elements of parametrized higher category theory