Book excerpt: Realization of the conventional Von Neumann architecture faces increasing challenges due to growing process variations, device reliability and power consumption. As an appealing architectural solution, brain-inspired neuromorphic computing has drawn a great deal of research interest due to its potential improved scalability and power efficiency, and better suitability in processing complex tasks. Moreover, inherit error resilience in neuromorphic computing allows remarkable power and energy savings by exploiting approximate computing. This dissertation focuses on a scalable and energy efficient neurocomputing architecture which leverages emerging memristor nanodevices and a novel approximate arithmetic for cognitive computing. First, brain-inspired digital neuromorphic processor (DNP) architecture with memristive synaptic crossbar is presented for large scale spiking neural networks. We leverage memristor nanodevices to build an N x N crossbar array to store not only multibit synaptic weight values but also the network configuration data with significantly reduced area cost. Additionally, the crossbar array is accessible both column- and row-wise to significantly expedite the synaptic weight update process for on-chip learning. The proposed digital pulse width modulator (PWM) readily creates a binary pulse with various durations to read and write the multilevel memristors with low cost. Our design integrates N digital leaky integrate-and-fire (LIF) silicon neurons to mimic their biological counterparts and the respective on-chip learning circuits for implementing spike timing dependent plasticity (STDP) learning rules. The proposed column based analog-to-digital conversion (ADC) scheme accumulates the pre-synaptic weights of a neuron efficiently and reduces silicon area by using only one shared arithmetic unit for processing LIF operations of all N neurons. With 256 silicon neurons, the learning circuits and 64K synapses, the power dissipation and area of our design are evaluated as 6.45 mW and 1.86 mm2, respectively, in a 90 nm CMOS technology. Furthermore, arithmetic computations contribute significantly to the overall processing time and power of the proposed architecture. In particular, addition and comparison operations represent 88.5% and 42.9% of processing time and power for digital LIF computation, respectively. Hence, by exploiting the built-in resilience of the presented neuromorphic architecture, we propose novel approximate adder and comparator designs to significantly reduce energy consumption with a very low error rate. The significantly improved error rate and critical path delay stem from a novel carry prediction technique that leverages the information from less significant input bits in a parallel manner. An error magnitude reduction scheme is proposed to further reduce amount of error once detected with low cost in the proposed adder design. Implemented in a commercial 90 nm CMOS process, it is shown that the proposed adder is up to 2.4x faster and 43% more energy efficient over traditional adders while having an error rate of only 0.18%. Additionally, the proposed comparator achieves an error rate of less than 0.1% and an energy reduction of up to 4.9x compared to the conventional ones. The proposed arithmetic has been adopted in a VLSI-based neuromorphic character recognition chip using unsupervised learning. The approximation errors of the proposed arithmetic units have been shown to have negligible impacts on the training process. Moreover, the energy saving of up to 66.5% over traditional arithmetic units is achieved for the neuromorphic chip with scaled supply levels. The electronic version of this dissertation is accessible from http://hdl.handle.net/1969.1/151721