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Quasinormal modes of a massive Dirac field were calculated for various staticblack hole backgrounds with the help of the WKB formula. These estimations,however, are rough and valid only for very small values of $\mu M$, where $M$and $\mu$ are mass of the black hole and field respectively. Thus, no accuratecalculations of massive Dirac modes are known even for the Schwarzschild blackhole and this is all the more so for the Kerr solution. Here we calculatequasinormal modes of a massive Dirac field in the Kerr background. We haveshown that the infinitely long-lived quasinormal modes (quasiresonances), whichexist for boson fields, appear also in the fermions' quasinormal spectrum. Twochiralities of massive fermions lead to an additional "fine structure" in thespectrum. We discuss the effect of this fine structure on the behavior ofquasiresonances and the stability. The analysis is also extended to a chargedmassive field in the Kerr-Newman background.

Quasinormal modes of massive fermions in Kerr spacetime: Long-lived modes and the fine structure