Artificial Intelligence has found many applications in the last decade due to increased computing power. Artificial Neural Networks are inspired in the brain structure and consist in the interconnection of artificial neurons through artificial synapses in the so-called Deep Neural Networks (DNNs). Training these systems requires huge amounts of data and, after the network is trained, it can recognize unforeseen data and provide useful information. As far as the training is concerned, we can distinguish between supervised and unsupervised learning. The former requires labelled data and is based on the iterative minimization of the output error using the stochastic gradient descent method followed by the recalculation of the strength of the synaptic connections (weights) with the backpropagation algorithm. On the other hand, unsupervised learning does not require data labeling and it is not based on explicit output error minimization. Conventional ANNs can function with supervised learning algorithms (perceptrons, multi-layer perceptrons, convolutional networks, etc.) but also with unsupervised learning rules (Kohonen networks, self-organizing maps, etc.). Besides, another type of neural networks are the so-called Spiking Neural Networks (SNNs) in which learning takes place through the superposition of voltage spikes launched by the neurons. Their behavior is much closer to the brain functioning mechanisms they can be used with supervised and unsupervised learning rules. Since learning and inference is based on short voltage spikes, energy efficiency improves substantially. Up to this moment, all these ANNs (spiking and conventional) have been implemented as software tools running on conventional computing units based on the von Neumann architecture. However, this approach reaches important limits due to the required computing power, physical size and energy consumption. This is particularly true for applications at the edge of the internet. Thus, there is an increasing interest in developing AI tools directly implemented in hardware for this type of applications. The first hardware demonstrations have been based on Complementary Metal-Oxide-Semiconductor (CMOS) circuits and specific communication protocols. However, to further increase training speed andenergy efficiency while reducing the system size, the combination of CMOS neuron circuits with memristor synapses is now being explored. It has also been pointed out that the short time non-volatility of some memristors may even allow fabricating purely memristive ANNs. The memristor is a new device (first demonstrated in solid-state in 2008) which behaves as a resistor with memory and which has been shown to have potentiation and depression properties similar to those of biological synapses. In this Special Issue, we explore the state of the art of neuromorphic circuits implementing neural networks with memristors for AI applications.