  We study the geometry of non-homogeneous horospherical varieties. These have
been classified by Pasquier and include the well-known odd symplectic
Grassmannians. We focus our study on quantum cohomology, with a view towards
Dubrovin's conjecture. In particular, we describe the cohomology groups of
these varieties as well as a Chevalley formula, and prove that many
Gromov-Witten invariants are enumerative. This enables us to prove that in many
cases the quantum cohomology is semisimple. We give a presentation of the
quantum cohomology ring for odd symplectic Grassmannians. The final section is
devoted to the derived categories of coherent sheaves on horospherical
varieties. We first discuss a general construction of exceptional bundles on
these varieties. We then study in detail the case of the horospherical variety
associated to the exceptional group $G_2$, and construct a full rectangular
Lefschetz exceptional collection in the derived category.
