  This paper is about the Fukaya category of a Fano hypersurface $X \subset
\mathbb{CP}^n$. Because these symplectic manifolds are monotone, both the
analysis and the algebra involved in the definition of the Fukaya category
simplify considerably. The first part of the paper is devoted to establishing
the main structures of the Fukaya category in the monotone case: the
closed-open string maps, weak proper Calabi-Yau structure, Abouzaid's
split-generation criterion, and their analogues when weak bounding cochains are
included. We then turn to computations of the Fukaya category of the
hypersurface $X$: we construct a configuration of monotone Lagrangian spheres
in $X$, and compute the associated disc potential. The result coincides with
the Hori-Vafa superpotential for the mirror of $X$ (up to a constant shift in
the Fano index $1$ case). As a consequence, we give a proof of Kontsevich's
homological mirror symmetry conjecture for $X$. We also explain how to extract
non-trivial information about Gromov-Witten invariants of $X$ from its Fukaya
category.

Symplectic Geometry