One important class of applications for the Internet of Things is related to the need to gain timely and continuous situational awareness, like smart cities, automated traffic control, or emergency and rescue operations. Events happening in the real-world need to be detected in real-time based on sensor data and other data sources. Complex Event Processing (CEP) is a technology to detect complex (or composite) events in data streams and has been successfully applied in high volume and high velocity applications like stock market analysis. However, these application domains faced only the challenge of high performance, while the Internet of Things and Mobile Big Data introduce a new set of challenges caused by mobility. This chapter aims to explain these challenges and to give an overview on how they are solved respectively how far state-of-the-art research has advanced to be useful to solve Mobile Big Data problems. At the infrastructure level the main challenge is to trade performance against resource consumption and energy efficiency and operator placement is the most dominant mechanism to address these problems. At the application and consumer level, mobile queries pose a new set of challenges for CEP related to continuously changing positions of consumers and data sources, and the need to adapt the query processing to these changes. Finally, proper methods and tools for systematical testing and reproducible performance evaluation for mobile distributed CEP are needed but not yet available.