Web search is an experience that naturally lends itself to recommendations, including query suggestions and related entities. In this article, we propose to recommend specific tasks to users, based on their search queries, such as planning a holiday trip or organizing a party. Specifically, we introduce the problem of query-based task recommendation and develop methods that combine well-established term-based ranking techniques with continuous semantic representations, including sentence representations from several transformer-based models. Using a purpose-built test collection, we find that our method is able to significantly outperform a strong text-based baseline. Further, we extend our approach to using a set of queries that all share the same underlying task, referred to as search mission, as input. The study is rounded off with a detailed feature and query analysis.