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This paper describes a motion-adaptive indexing scheme for efficient evaluation of moving continual queries (MCQs) over moving objects. It uses the concept of motion-sensitive bounding boxes (MSBs) to model moving objects and moving queries. These bounding boxes automatically adapt their sizes to the dynamic motion behaviors of individual objects. Instead of indexing frequently changing object positions, we index less frequently changing object and query MSBs, where updates to the bounding boxes are needed only when objects and queries move across the boundaries of their boxes. This helps decrease the number of updates to the indexes. More importantly, we use predictive query results to optimistically precalculate query results, decreasing the number of searches on the indexes. Motion-sensitive bounding boxes are used to incrementally update the predictive query results. Furthermore, we introduce the concepts of guaranteed safe radius and optimistic safe radius to extend our motion-adaptive indexing scheme to evaluating moving continual k-nearest neighbor (kNN) queries. Our experiments show that the proposed motion-adaptive indexing scheme is efficient for the evaluation of both moving continual range queries and moving continual kNN queries.