  Gromov-Witten theory is used to define an enumerative geometry of curves in
Calabi-Yau 5-folds. We find recursions for meeting numbers of genus 0 curves,
and we determine the contributions of moving multiple covers of genus 0 curves
to the genus 1 Gromov-Witten invariants. The resulting invariants, conjectured
to be integral, are analogous to the previously defined BPS counts for
Calabi-Yau 3 and 4-folds. We comment on the situation in higher dimensions
where new issues arise.
  Two main examples are considered: the local Calabi-Yau P^2 with balanced
normal bundle 3O(-1) and the compact Calabi-Yau hypersurface X_7 in P^6. In the
former case, a closed form for our integer invariants has been conjectured by
G. Martin. In the latter case, we recover in low degrees the classical
enumeration of elliptic curves by Ellingsrud and Stromme.
