Desktop Grids and Volunteer Computing Systems (DGVCSs) are approaches of distributed systems aimed at the provision of large scale computing infrastructures to support eScience project, by taking advantage of non-dedicated resources, most of them desktop computers presenting low levels of use. Those computers are available through Internet or Intranet environments, have partial availability, are highly heterogeneous, and are part of independent administrative domains. Currently, the idle computational capabilities of millions of volunteer distributed computing resources are exploited without affecting the quality of service perceived by the end users. This chapter presents a comprehensive state of the art of DGVCSs, providing a global picture of the main solutions and research trends in DGVCSs. It will discuss the evolution of such systems by analyzing representative examples. We identify the main characteristics for DGVCSs, and we introduce a new taxonomy to categorize these projects to make their study easier.