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A number of stability results for general systems of linear and nonlinear partial differential equations that describe distributed parameter networks and systems are obtained by means of Liapunov's direct method. Conditions for the asymptotic stability of the null solution of linear time-invariant distributed parameter systems are given. This result is then extended to a class of distributed parameter systems consisting of a linear system and a zero-memory nonlinear elment described by a set of Lurie-like partial differential equations. Bounded input-bounded output stability for the above two classes of systems is established by extension of a result of Goldwyn, Chao, and Chang for lumped parameter systems. Finally, it is shown how Malkin's theorem on the stability under persistent disturbances can be extended to provide a solution to the dynamic range problem for systems described by a general set of nonlinear partial differential equations.