Hermite and Laguerre β ensembles are two of the most well studied models in random matrix theory with the special cases β=1,2,4 corresponding to eigenvalue distributions of classical Gaussian and Wishart ensembles.

Using tridiagonal matrix models for these ensembles introduced by Dumitriu and Edelman, Ramirez, Rider and Virag established Tracy-Widom β scaling limits for the largest eigenvalues. They also showed that the upper tail decays of the Tracy-Widom β distribution decays like exp(-2βt^{3/2}/30 while the lower tail decays like exp(-βt^3/24).  I shall describe some recent work establishing sharp tail estimates in the pre-limiting models up to 1+o(1) error at the level of exponents, improving earlier results by of Ledoux and Rider. I shall also describe a number of applications of these estimates to exactly solvable planar last passage percolation, which are related to the classical β ensembles via some remarkable distributional identities.