In the first part of this chapter, based on the monotone dynamical systems theory, necessary and sufficient delay-independent exponential stability conditions are formulated for 2-D nonlinear systems in the Roesser model with time-varying delays. Applications to the stability analysis of positive 2-D systems with delays are also obtained. In the second part, the design problem of positive observers for 2-D positive linear systems with delays is investigated. Extended Luenberger-type observers that generate nonnegative asymptotic estimations of state vectors are constructed for 2-D positive time-delay systems described by the Roesser model and Fornasini–Marchesini local state-space model. A reduced-order functional state observer that is an asymptotic approximation of a given linear functional of the state vector is also taken into consideration for 2-D positive time-delay systems in the Roesser model. Necessary and sufficient delay-independent design conditions are derived using linear programming approach. The efficacy of the design method is demonstrated by numerical examples.