We present a fully memristive spiking neural network (MSNN) consisting of novel memristive neurons named memristive integrate-and-fire (MIF) and memristive synapses implemented an unsupervised Spike Timing Dependent Plasticity learning rule. 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. The use of the MIF neuron model in large scale MSNNs to emulate brain-inspired computing is demonstrated by replicating biological memory retrieval mechanisms with the proposed MSNN, and achieving 97.5% accuracy in a 4-pattern recognition problem using a discriminative MSNN.