Densest subgraph discovery (DSD) is a fundamental problem in graph mining. It has been studied for decades, and is widely used in various areas, including network science, biological analysis, and graph databases. Given a graph G, DSD aims to find a subgraph D of G with the highest density (e.g., the number of edges over the number of vertices in D). Because DSD is difficult to solve, we propose a new solution paradigm in this paper. Our main observation is that a densest subgraph can be accurately found through a k-core (a kind of dense subgraph of G), with theoretical guarantees. Based on this intuition, we develop efficient exact and approximation solutions for DSD. Moreover, our solutions are able to find the densest subgraphs for a wide range of graph density definitions, including clique-based and general pattern-based density. We have performed extensive experimental evaluation on eleven real datasets. Our results show that our algorithms are up to four orders of magnitude faster than existing approaches.