Abstract: In this paper, we discuss the so-called time-fractional telegraph equation. It is a generalization of the classical telegraph equation in case the first- and two-order time derivatives are replaced with Caputo derivatives of order \alpha\in(\frac12,1, 2\alpha\in(1,2. By using the spatial finite sine and cosine transform, and the temporal Laplace transform, the existence of the analytic solutions of its initial-boundary problems in a boundedspace domain with Dirichlet and Neumann boundary conditions is derived. The analytic solutions are given in the form of series of the Mittag-Leffler functions.