In today's nanoscale era, scaling down to even smaller feature sizes poses a significant challenge in the device fabrication, the circuit, and the system design and integration. On the other hand, nanoscale technology has also led to novel materials and devices with unique properties. The memristor is one such emergent nanoscale device that exhibits non-linear current-voltage characteristics and has an inherent memory property, i.e., its current state depends on the past. Both the non-linear and the memory property of memristors have the potential to enable solving spatial and temporal pattern recognition tasks in radically different ways from traditional binary transistor-based technology. The goal of this thesis is to explore the use of memristors in a novel computing paradigm called "Reservoir Computing" (RC). RC is a new paradigm that belongs to the class of artificial recurrent neural networks (RNN). However, it architecturally differs from the traditional RNN techniques in that the pre-processor (i.e., the reservoir) is made up of random recurrently connected non-linear elements. Learning is only implemented at the readout (i.e., the output) layer, which reduces the learning complexity significantly. To the best of our knowledge, memristors have never been used as reservoir components. We use pattern recognition and classification tasks as benchmark problems. Real world applications associated with these tasks include process control, speech recognition, and signal processing. We have built a software framework, RCspice (Reservoir Computing Simulation Program with Integrated Circuit Emphasis), for this purpose. The framework allows to create random memristor networks, to simulate and evaluate them in Ngspice, and to train the readout layer by means of Genetic Algorithms (GA). We have explored reservoir-related parameters, such as the network connectivity and the reservoir size along with the GA parameters. Our results show that we are able to efficiently and robustly classify time-series patterns using memristor-based dynamical reservoirs. This presents an important step towards computing with memristor-based nanoscale systems.