In this paper we address the problem of scalable, native and adaptive query processing over Linked Stream Data integrated with Linked Data. Linked Stream Data consists of data generated by stream sources, e.g., sensors, enriched with semantic descriptions, following the standards proposed for Linked Data. This enables the integration of stream data with Linked Data collections and facilitates a wide range of novel applications. Currently available systems use a “black box” approach which delegates the processing to other engines such as stream/event processing engines and SPARQL query processors by translating to their provided languages. As the experimental results described in this paper show, the need for query translation and data transformation, as well as the lack of full control over the query execution, pose major drawbacks in terms of efficiency. To remedy these drawbacks, we present CQELS (Continuous Query Evaluation over Linked Streams), a native and adaptive query processor for unified query processing over Linked Stream Data and Linked Data. In contrast to the existing systems, CQELS uses a “white box” approach and implements the required query operators natively to avoid the overhead and limitations of closed system regimes. CQELS provides a flexible query execution framework with the query processor dynamically adapting to the changes in the input data. During query execution, it continuously reorders operators according to some heuristics to achieve improved query execution in terms of delay and complexity. Moreover, external disk access on large Linked Data collections is reduced with the use of data encoding and caching of intermediate query results. To demonstrate the efficiency of our approach, we present extensive experimental performance evaluations in terms of query execution time, under varied query types, dataset sizes, and number of parallel queries. These results show that CQELS outperforms related approaches by orders of magnitude.