Domain adaptation aims to exploit the supervision knowledge in a source domain for learning prediction models in a target domain. In this article, we propose a novel representation learning-based domain adaptation method, i.e., neural embedding matching (NEM) method, to transfer information from the source domain to the target domain where labeled data is scarce. The proposed approach induces an intermediate common representation space for both domains with a neural network model while matching the embedding of data from the two domains in this common representation space. The embedding matching is based on the fundamental assumptions that a cross-domain pair of instances will be close to each other in the embedding space if they belong to the same class category, and the local geometry property of the data can be maintained in the embedding space. The assumptions are encoded via objectives of metric learning and graph embedding techniques to regularize and learn the semisupervised neural embedding model. We also provide a generalization bound analysis for the proposed domain adaptation method. Meanwhile, a progressive learning strategy is proposed and it improves the generalization ability of the neural network gradually. Experiments are conducted on a number of benchmark data sets and the results demonstrate that the proposed method outperforms several state-of-the-art domain adaptation methods and the progressive learning strategy is promising.