Many web-search queries serve as the beginning of an exploration of an unknown space of information, rather than looking for a specific web page. To answer such queries effec- tively, the search engine should attempt to organize the space of relevant information in a way that facilitates exploration. We describe the Aspector system that computes aspects for a given query. Each aspect is a set of search queries that together represent a distinct information need relevant to the original search query. To serve as an effective means to explore the space, Aspector computes aspects that are orthogonal to each other and to have high combined coverage. Aspector combines two sources of information to compute aspects. We discover candidate aspects by analyzing query logs, and cluster them to eliminate redundancies. We then use a mass-collaboration knowledge base (e.g., Wikipedia) to compute candidate aspects for queries that occur less frequently and to group together aspects that are likely to be "semantically" related. We present a user study that indicates that the aspects we compute are rated favorably against three competing alternatives -related searches proposed by Google, cluster labels assigned by the Clusty search engine, and navigational searches proposed by Bing.