Semi-supervised learning is drawing increasing attention in the era of big data, as the gap between the abundance of cheap, automatically collected unlabeled data and the scarcity of labeled data that are laborious and expensive to obtain is dramatically increasing. In this paper, we first introduce a unified view of density-based clustering algorithms. We then build upon this view and bridge the areas of semi-supervised clustering and classification under a common umbrella of density-based techniques. We show that there are close relations between density-based clustering algorithms and the graph-based approach for transductive classification. These relations are then used as a basis for a new framework for semi-supervised classification based on building-blocks from density-based clustering. This framework is not only efficient and effective, but it is also statistically sound. In addition, we generalize the core algorithm in our framework, HDBSCAN*, so that it can also perform semi-supervised clustering by directly taking advantage of any fraction of labeled data that may be available. Experimental results on a large collection of datasets show the advantages of the proposed approach both for semi-supervised classification as well as for semi-supervised clustering.