The formalism of P systems was introduced by Gheorghe Paun under the name of membrane systems about ten years ago, yet just recently new derivation modes and halting conditions have been proposed. For developing comparable results, a formal description of their functioning, in particular, of a derivation step was necessary. A general formal framework of networks of cells was introduced to capture most of the essential features of static (tissue) P systems and to define their functioning in a formal way. In this book, the formal framework is extended by some new examples as well as by new halting methods and derivation modes. With non-halting computations, we even can go beyond Turing computability. P systems with non-co-operative rules and the maximally parallel derivation mode characterize families of sets of natural numbers defined by Lindenmayer systems when considering the results obtained in a specific output cell in every derivation step during any computation. With the new derivation mode of 1-restricted minimal parallelism, we are able to model spiking neural P systems as well as purely catalytic P systems as networks of cells.