Summary: We establish necessary and sufficient conditions on bounded linear operators for which generalized Weyl’s theorem, or the generalized \(a\)-Weyl theorem, holds. We also consider generalized Weyl’s theorems in the framework of polaroid operators and obtain improvements of some results recently established by R. E. Curto and Y. M. Han [J. Math. Anal. Appl. 336, No. 2, 1424–1442 (2007; Zbl 1131.47003)] and H. Zguitti [J. Math. Anal. Appl. 316, No. 1, 373–381 (2006; Zbl 1101.47002)].