We seek to (i) characterize the learning architectures exploited in biological neural networks for training on very few samples, and (ii) port these algorithmic structures to a machine learning context. The Moth Olfactory Network is among the simplest biological neural systems that can learn, and its architecture includes key structural elements and mechanisms widespread in biological neural nets, such as cascaded networks, competitive inhibition, high intrinsic noise, sparsity, reward mechanisms, and Hebbian plasticity. These structural biological elements, in combination, enable rapid learning. MothNet is a computational model of the Moth Olfactory Network, closely aligned with the moth's known biophysics and with in vivo electrode data collected from moths learning new odors. We assign this model the task of learning to read the MNIST digits. We show that MothNet successfully learns to read given very few training samples (1 to 10 samples per class). In this few-samples regime, it outperforms standard machine learning methods such as nearest-neighbors, support-vector machines, and neural networks (NNs), and matches specialized one-shot transfer-learning methods but without the need for pre-training. The MothNet architecture illustrates how algorithmic structures derived from biological brains can be used to build alternative NNs that may avoid some of the learning rate limitations of current engineered NNs.