  In this paper we characterize irreducible generic representations of
$\SO_{2n+1}(k)$ where $k$ is a $p$-adic field) by means of twisted local gamma
factors (the Local Converse Theorem). As applications, we prove that two
irreducible generic cuspidal automorphic representations of $\SO_{2n+1}({\Bbb
A})$ (where ${\Bbb A}$ is the ring of adeles of a number field) are equivalent
if their local components are equivalent at almost all local places (the
Rigidity Theorem);and prove the Local Langlands Reciprocity Conjecture for
generic supercuspidal representations of $\SO_{2n+1}(k)$.
