Sampled-data observers/controllers under the sampled in space and time measurements were suggested in the past for parabolic systems. In the present paper, for the first time, a sampled-data observer is constructed for a hyperbolic system governed by 1D semilinear wave equation with either viscous or boundary damping. The measurements are sampled in space and time. Sufficient conditions for the exponential stability of the estimation error are derived by using the time-delay approach to sampled-data control and appropriate Lyapunov-Krasovskii functionals. The dual sampled data controller problems are formulated. Numerical examples including observer design for unstable damped sine-Gordon equation illustrate the efficiency of the method. (C) 2019 Elsevier Ltd. All rights reserved.