A duoidal category is a category equipped with two monoidal structures in
which one is (op)lax monoidal with respect to the other. In this paper we
introduce duoidal $\infty$-categories which are counterparts of duoidal
categories in the setting of $\infty$-categories. There are three kinds of
functors between duoidal $\infty$-categories, which are called bilax, double
lax, and double oplax monoidal functors. We make three formulations of
$\infty$-categories of duoidal $\infty$-categories according to which functors
we take. Furthermore, corresponding to the three kinds of functors, we define
bimonoids, double monoids, and double comonoids in duoidal $\infty$-categories.