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We present a fully memristive spiking neural network (MSNN) consisting of physically-realizable memristive neurons and memristive synapses to implement an unsupervised Spiking Time Dependent Plasticity (STDP) learning rule. The system is fully memristive in that both neuronal and synaptic dynamics can be realized by using memristors. The neuron is implemented using the SPICE-level memristive integrate-and-fire (MIF) model, which consists of a minimal number of circuit elements necessary to achieve distinct depolarization, hyperpolarization, and repolarization voltage waveforms. The proposed MSNN uniquely implements STDP learning by using cumulative weight changes in memristive synapses from the voltage waveform changes across the synapses, which arise from the presynaptic and postsynaptic spiking voltage signals during the training process. Two types of MSNN architectures are investigated: 1) a biologically plausible memory retrieval system, and 2) a multi-class classification system. Our circuit simulation results verify the MSNN's unsupervised learning efficacy by replicating biological memory retrieval mechanisms, and achieving 97.5% accuracy in a 4-pattern recognition problem in a large scale discriminative MSNN.