  We study discrete $\beta$-ensembles as introduced in [17]. We obtain rigidity
estimates on the particle locations, i.e. with high probability, the particles
are close to their classical locations with an optimal error estimate. We prove
the edge universality of the discrete $\beta$-ensembles, i.e. for $\beta\geq
1$, the distribution of extreme particles converges to the Tracy-Widom $\beta$
distribution. As far as we know, this is the first proof of general Tracy-Widom
$\beta$ distributions in the discrete setting. A special case of our main
results implies that under the Jack deformation of the Plancherel measure, the
distribution of the lengths of the first few rows in Young diagrams, converges
to the Tracy-Widom $\beta$ distribution, which answers an open problem in [39].
Our proof relies on Nekrasov's (or loop) equations, a multiscale analysis and a
comparison argument with continuous $\beta$-ensembles.
