Recent advances in types and extent of medical imaging technologies has led to proliferation of multimodal quantitative imaging data in cancer. Quantitative medical imaging data refer to numerical representations derived from medical imaging technologies, such as radiology and pathology imaging, that can be used to assess and quantify characteristics of diseases, especially cancer. The use of such data in both clinical and research setting enables precise quantifications and analyses of tumor characteristics that can facilitate objective evaluation of disease progression, response to therapy, and prognosis. The scale and size of these imaging biomarkers is vast and presents several analytical and computational challenges that range from high-dimensionality to complex structural correlation patterns. In this review article, we summarize some state-of-the-art statistical methods developed for quantitative medical imaging data ranging from topological, functional and shape data analyses to spatial process models. We delve into common imaging biomarkers with a focus on radiology and pathology imaging in cancer, address the analytical questions and challenges they present, and highlight the innovative statistical and machine learning models that have been developed to answer relevant scientific and clinical questions. We also outline some emerging and open problems in this area for future explorations.