  Jacobi forms can be considered as vector valued modular forms, and Jacobi
forms of critical weight correspond to vector valued modular forms of weight
$\frac12$. Since the only modular forms of weight $\frac12$ on congruence
subgroups of $\SL$ are theta series the theory of Jacobi forms of critical
weight is intimately related to the theory of Weil representations of finite
quadratic modules. This article explains this relation in detail, gives an
account of various facts about Weil representations which are useful in this
context, and it gives some applications of the theory developed herein by
proving various vanishing theorems and by proving a conjecture on Jacobi forms
of weight one on $\SL$ with character.
