Online learning (OL) from data streams is an emerging area of research that encompasses numerous challenges from stream processing, machine learning, and networking. Stream-processing platforms, such as Apache Kafka and Flink, have basic extensions for the training of Artificial Neural Networks (ANNs) in a stream-processing pipeline. However, these extensions were not designed to train ANNs in real-time, and they suffer from performance and scalability issues when doing so. This paper presents TensAIR, the first OL system for training ANNs in real time. TensAIR achieves remarkable performance and scalability by using a decentralized and asynchronous architecture to train ANN models (either freshly initialized or pre-trained) via DASGD (decentralized and asynchronous stochastic gradient descent). We empirically demonstrate that TensAIR achieves a nearly linear scale-out performance in terms of (1) the number of worker nodes deployed in the network, and (2) the throughput at which the data batches arrive at the dataflow operators. We depict the versatility of TensAIR by investigating both sparse (word embedding) and dense (image classification) use cases, for which TensAIR achieved from 6 to 116 times higher sustainable throughput rates than state-of-the-art systems for training ANN in a stream-processing pipeline.