We consider the Goldstein-Kac telegraph process X(t), t > 0, on the real line R1 performed by the random motion at finite speed c and controlled by a homogeneous Poisson process of rate  > 0. Using a formula for the moment function μ2k(t) of X(t) we study its asymptotic behaviour, as c,  and t vary in different ways. Explicit asymptotic formulas for μ2k(t), as k → ∞, are derived and numerical comparison of their effectiveness is given. We also prove that the moments μ2k(t) for arbitrary fixed t > 0 satisfy the Carleman condition and, therefore, the distribution of the telegraph process is completely determined by its moments. Thus, the moment problem is completely solved for the telegraph process X(t). We obtain an explicit formula for the Laplace transform of μ2k(t) and give a derivation of the the moment generating function based on direct calculations. A formula for the semi-invariants of X(t) is also presented.