Spiking neural networks (SNNs) have emerged as a promising brain inspired neuromorphic-computing paradigm for cognitive system design due to their inherent event-driven processing capability. The fully connected (FC) shallow SNNs typically used for pattern recognition require large number of trainable parameters to achieve competitive classification accuracy. In this paper, we propose a deep spiking convolutional neural network (SpiCNN) composed of a hierarchy of stacked convolutional layers followed by a spatial-pooling layer and a final FC layer. The network is populated with biologically plausible leaky-integrate-and-fire (LIF) neurons interconnected by shared synaptic weight kernels. We train convolutional kernels layer-by-layer in an unsupervised manner using spike-timingdependent plasticity (STDP) that enables them to self-learn characteristic features making up the input patterns. In order to further improve the feature learning efficiency, we propose using smaller 3×3 kernels trained using STDP-based synaptic weight updates performed over a mini-batch of input patterns. Our deep SpiCNN, consisting of two convolutional layers trained using the unsupervised convolutional STDP learning methodology, achieved classification accuracies of 91.1% and 97.6%, respectively, for inferring handwritten digits from the MNIST data set and a subset of natural images from the Caltech data set.