In the field of membrane computing, a great attention is traditionally paid to the results demonstrating a theoretical possibility to solve NP-complete problems in polynomial time by means of various models of P systems. A bit less common is the fact that almost all models of P systems with this capability are actually stronger: some of them are able to solve PSPACEcomplete problems in polynomial time, while others have been recently shown to characterize the class # (which is conjectured to be

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... strictly included in PSPACE). A large part of these results has appeared quite recently. In this survey, we focus on strong models of membrane systems which have the potential to solve hard problems belonging to classes containing NP. These include P systems with active membranes, P systems with proteins on membranes and tissue P systems, as well as P systems with symport/antiport and membrane division and, finally, spiking neural P systems. We provide a survey of computational complexity results of these membrane models, pointing out some features providing them with their computational strength. We also mention a sequence of open problems related to these topics.