Bounds on the exponential decay of generalized eigenfunctions of bounded and unbounded selfadjoint Jacobi matrices in ℓ 2(ℕ) are established. Two cases are considered separately and lead to different results: (i) the case in which the spectral parameter lies in a general gap of the spectrum of the Jacobi matrix and (ii) the case of a lower semi-bounded Jacobi matrix with values of the spectral parameter below the spectrum. It is demonstrated by examples that both results are sharp.We apply these results to obtain a "many barriers-type" criterion for the existence of square-summable generalized eigenfunctions of an unbounded Jacobimatrix at almost every value of the spectral parameter in suitable open sets. In particular, this leads to examples of unbounded Jacobi matrices with a spectral mobility edge, i.e., a transition from purely absolutely continuous spectrum to dense pure point spectrum. © 2008 The Author Published by Oxford University Press. All rights reserved.