Every day a large number of Earth observation (EO) spaceborne and airborne sensors from many different countries provide a massive amount of remotely sensed data. Those data are used for different applications, such as natural hazard monitoring, global climate change, urban planning, etc. The applications are data driven and mostly interdisciplinary. Based on this it can truly be stated that we are now living in the age of big remote sensing data. Furthermore, these data are becoming an economic asset and a new important resource in many applications. In this paper, we specifically analyze the challenges and opportunities that big data bring in the context of remote sensing applications. Our focus is to analyze what exactly does big data mean in remote sensing applications and how can big data provide added value in this context. Furthermore, this paper describes the most challenging issues in managing, processing, and efficient exploitation of big data for remote sensing problems. In order to illustrate the aforementioned aspects, two case studies discussing the use of big data in remote sensing are demonstrated. In the first test case, big data are used to automatically detect marine oil spills using a large archive of remote sensing data.