This thesis presents new results in the field of state estimation based on the theory of positive systems. It is composed of two separate parts. The first one studies the problem of positive observer design for positive systems. The second one which deals with robust state estimation through the design of interval observers, is at the core of our work.We begin our thesis by proposing the design of a nonlinear positive observer for discrete-time positive time-varying linear systems based on the use of generalized polar coordinates in the positive orthant. For positive systems, a natural requirement is that the observers should provide state estimates that are also non-negative so they can be given a physical meaning at all times. The idea underlying the method is that first, the direction of the true state is correctly estimated in the projective space thanks to the Hilbert metric and then very mild assumptions on the output map allow to reconstruct the norm of the state. The convergence rate can be controlled.Later, the thesis is continued by studying the so-called interval observers for different families of dynamic systems in continuous-time, in discrete-time and also in a context "continuous-discrete" (i.e. a class of continuous-time systems with discrete-time measurements). Interval observers are dynamic extensions giving estimates of the solution of a system in the presence of various type of disturbances through two outputs giving an upper and a lower bound for the solution. Thanks to interval observers, one can construct control laws which stabilize the considered systems.