This chapter proposes some non-trivial extensions of the classical highgain observer designs for finite-dimensional nonlinear systems to some classes of infinite-dimensional systems, written as triangular systems of coupled first-order hyperbolic partial differential equations (PDEs), where an observation of one coordinate of the state only is considered as the system's output. These forms may include some epidemic models and tubular chemical reactors. To deal with this problem, depending on the number of distinct velocities of the hyperbolic system, direct and indirect observer designs are proposed. We first show intuitively how direct observer design can be applied to quasilinear partial integro-differential hyperbolic systems of balance laws with a single velocity, as a natural extension of the finitedimensional case. We then introduce an indirect approach for systems with distinct velocities (up to three velocities), where an infinite-dimensional state transformation first maps the system into a suitable systems of Partial Differential Equations (PDEs) and the convergence of the observer is subsequently exhibited in appropriate norms. This indirect approach leads to the use of spatial derivatives of the output in the observer dynamics.