Due to a variety of reasons including data randomness and incompleteness, noise, privacy, etc., uncertainty is inherent in many important applications, such as location-based services (LBS), sensor network monitoring, and radio-frequency identification (RFID). Recently, considerable research efforts have been devoted into the field of uncertainty-aware spatial query processing such that the uncertainty of the data can be effectively and efficiently tackled. In this paper, we study the problem of finding top $k$ <alternatives><inline-graphic xlink:type="simple" xlink:href="zhang-ieq2-2457899.gif"/></alternatives> most influential facilities over a set of uncertain objects, which is an important and fundamental spatial query in the above applications. Based on the maximal utility principle, we propose a new ranking model to identify the top $k$<alternatives> <inline-graphic xlink:type="simple" xlink:href="zhang-ieq3-2457899.gif"/></alternatives> most influential facilities, which carefully captures influence of facilities on the uncertain objects. By utilizing two uncertain object indexing techniques, $R$<alternatives> <inline-graphic xlink:type="simple" xlink:href="zhang-ieq4-2457899.gif"/></alternatives>-tree and $U$<alternatives><inline-graphic xlink:type="simple" xlink:href="zhang-ieq5-2457899.gif"/></alternatives> -Quadtree, effective and efficient algorithms are proposed following the filtering and verification paradigm, which significantly improves the performance of the algorithms in terms of CPU and I/O costs. To effectively support uncertain objects with a large number of instances, we also develop randomized algorithms with accuracy guarantee. Then, a hybrid algorithm is devised which effectively combines the randomized and exact algorithms. Comprehensive experiments on real datasets demonstrate the effectiveness and efficiency of our techniques.