As feature-size scaling and "Moore's Law" in integrated CMOS circuits further slows down, attention is shifting to computing by non-von Neumann and non-Boolean computing models. Reservoir computing (RC) is a new computing paradigm that allows to harness the intrinsic dynamics of a "reservoir" to perform useful computations. The reservoir, or compute core, must only provide sufficiently rich dynamics that are then mapped onto a low-dimensional space by an readout layer. One of the key advantages of this approach is that only the readout layer needs to be adapted to perform the desired computation. The reservoir itself remains unchanged. In this paper we use for the first time memristive components as reservoir building blocks that are assembled into device networks. Memristive components are particularly interesting for this purpose because of their non-linear and memory characteristics. In addition, they can be integrated very densely and provide rich dynamics with a few components only. We use pattern recognition and associative memory tasks to illustrate the memristive reservoir computing approach. For that purpose, we have built a software framework that allows to create valid memristor networks, to simulate and evaluate them in Ngspice, and to train the readout layer by means of a Genetic Algorithm (GA). Our results show that we can efficiently and robustly classify temporal patterns. The approach presents a promising new computing paradigm that harnesses the non-linear, time-dependent, and highly-variable properties of current memristive components for solving computational tasks.