Starting in the early 2000s, there has been a steadily growing interest in random finite sets in the research community worldwide. Since then, the theory of random finite sets has been successfully used in a wide variety of scientific fields on applications such as radar and sonar systems, air traffic control and navigation, telecommunications, medicine, audio and image processing, visual tracking in videos, robotics, agriculture, and forestry. The first main contribution of this thesis is to provide a systematic, detailed, and rigorous introduction to the theory of random finite sets that can serve as an entry point for readers with no prior exposure to this field. In the second main contribution, we apply the theory of random finite sets to the problem of estimating the states of an unknown and random number of objects based on image observations. We investigate a scenario where two independent sensors record partly overlapping images of a bigger scene and derive an estimator based on the posterior probability hypothesis density. We show how the estimation performance can be improved by exchanging information between the sensors over the case where each sensor calculates the estimates separately. Furthermore, we propose a novel algorithm to solve this estimation problem and demonstrate the algorithm-s performance in simulated scenarios.