  We define Gromov--Witten invariants of exploded manifolds. The technical
heart of this paper is a construction of a virtual fundamental class $[\mathcal
K]$ of any Kuranishi category $\mathcal K$ (which is a simplified, more general
version of an embedded Kuranishi structure.) We also show how to integrate
differential forms over $[\mathcal K]$ to obtain numerical invariants, and push
forward differential forms from $\mathcal K$ over suitable evaluation maps. We
show that such invariants are independent of any choices, and are compatible
with pullbacks, products, and tropical completion of Kuranishi categories.
  In the case of a compact symplectic manifold, this gives an alternative
construction of Gromov--Witten invariants, including gravitational descendants.
