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We prove that for any $0 < s < 1/2$, the Benjamin--Ono equation on the torusis globally in time $C^0-$well-posed on the Sobolev space $H^{-s}(\T, \R)$,inthe sense that the solution map, which is known to be defined for smooth data,continuously extends to $H^{-s}(\T,\R)$. The solution map does not extendcontinuously to $H^{-s}(\T, \R)$ with $s > 1/2$. Hence the critical Sobolevexponent $s_c=-1/2$ of the Benjamin--Ono equation is the threshold forwell-posedness on the torus. The obtained solutions are almost periodic intime. Furthermore, we prove that the traveling wave solutions of theBenjamin--Ono equation on the torus are orbitally stable in $H^{-s}(\T,\R)$ forany $0\le s<1/2$.

On the flow map of the Benjamin-Ono equation on the torus