Hybrid PET/MR systems have rapidly progressed from the prototype stage to systems that are increasingly being used in the clinics. This review provides an overview of developments in hybrid PET/MR systems and summarizes the current state of the art in PET/MR instrumentation, correction techniques, and data analysis. The strong magnetic field requires considerable changes in the manner by which PET images are acquired and has led, among others, to the development of new PET detectors, such as silicon photomultipliers. During more than a decade of active PET/MR development, several system designs have been described. The technical background of combined PET/MR systems is explained and related challenges are discussed. The necessity for PET attenuation correction required new methods based on MR data. Therefore, an overview of recent developments in this field is provided. Furthermore, MR-based motion correction techniques for PET are discussed, as integrated PET/MR systems provide a platform for measuring motion with high temporal resolution without additional instrumentation. The MR component in PET/MR systems can provide functional information about disease processes or brain function alongside anatomic images. Against this background, we point out new opportunities for data analysis in this new field of multimodal molecular imaging.