We will survey the known bounds on the eigenvalues of the Up Hecke operator appearing in various eigenvarieties. This will cover the speaker's work giving the polynomial growth rate of the associated Newton polygon for Chenevier's eigenvarieties for definite unitary groups of all ranks, as well as a theorem of Liu-Wan-Xiao and conjecture of Bergdall-Pollack giving more specific information for dimension-1 eigenvarieties.