  Certain Feynman integrals are associated to Calabi-Yau geometries. We
demonstrate how these integrals can be computed with the method of differential
equations. The four-loop equal-mass banana integral is the simplest Feynman
integral whose geometry is a non-trivial Calabi-Yau manifold. We show that its
differential equation can be cast into an $\varepsilon$-factorised form. This
allows us to obtain the solution to any desired order in the dimensional
regularisation parameter $\varepsilon$. The method generalises to other
Calabi-Yau Feynman integrals. Our calculation also shows that the four-loop
banana integral is only minimally more complicated than the corresponding
Feynman integrals at two or three loops.
