  In this paper we consider Jacobi forms of half-integral index for any
positive definite lattice L (classical Jacobi forms from the book of Eichler
and Zagier correspond to the lattice A_1=<2>). We give a lot of examples of
Jacobi forms of singular and critical weights for root systems using Jacobi
theta-series. We describe the Jacobi lifting for Jacobi forms of half-integral
indices. In some case it gives additive lifting construction of new reflective
modular forms.
