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We study the problem of processing subgraph queries on a database that consists of a set of graphs. The answer to a subgraph query is the set of graphs in the database that are supergraphs of the query. In this article, we propose an efficient index, FG*-index, to solve this problem. The cost of processing a subgraph query using most existing indexes mainly consists of two parts: the index probing cost and the candidate verification cost. Index probing is to find the query in the index, or to find the graphs from which we can generate a candidate answer set for the query. Candidate verification is to test whether each graph in the candidate set is indeed a supergraph of the query. We design FG*-index to minimize these two costs as follows. FG*-index consists of three components: the FG-index, the feature-index, and the FAQ-index. First, the FG-index employs the concept of Frequent subGraph (FG) to allow the set of queries that are FGs to be answered without candidate verification. We call this set of queries FG-queries. We can enlarge the set of FG-queries so that more queries can be answered without candidate verification; however, a larger set of FG-queries implies a larger FG-index and hence the index probing cost also increases. We propose the feature-index to reduce the index probing cost. The feature-index uses features to filter false results that are matched in the FG-index, so that we can quickly find the truly matching graphs for a query. For processing non-FG-queries, we propose the FAQ-index, which is dynamically constructed from the set of Frequently Asked non-FG-Queries (FAQs). Using the FAQ-index, verification is not required for processing FAQs and only a small number of candidates need to be verified for processing non-FG-queries that are not frequently asked. Finally, a comprehensive set of experiments verifies that query processing using FG*-index is up to orders of magnitude more efficient than state-of-the-art indexes and it is also more scalable.