Combining quantitative MR (qMR) with PET synergistically provides perfectly registered, complementary physiological biomarkers. Our pioneering work during the initial funding cycle, Y1-4, of our P41 in quantitative PET/MR, has provided solutions to some of the most vexing problems affecting PET/MR, including reliable and accurate corrections for attenuation correction and involuntary subject motion. These techniques achieve PET resolution close to the intrinsic resolution of the scanner and enable a range of novel PET/MR applications. In this renewal application, we will develop and deliver unique PET/MR imaging technologies that not only enable physiological high- impact novel PET/MR but will also benefit standalone MR research. We focus our efforts on multi- parametric qMR techniques by leveraging our recent breakthroughs on subspace/manifold modeling, MR-physics informed deep learning (DL), and multi-contrast ultra-fast MR sequences. Our proposed research focuses on developing and delivering new PET/MR imaging techniques along three axes: i) multi-parametric, free-running, cardiac MRI method for motion corrected whole-heart membrane potential imaging using PET/MR; ii) high resolution MRSI for multi-modality, multi-parametric, molecular imaging using PET/MR; and iii) ultra-fast, multi-component quantitative MR using deep learning and MR physics modeling. We will validate the proposed methods in TR&D3 PET/MR axes of research and through collaborations with our Collaborative Projects. We will leverage the DL methodologies proposed in TR&D2 to further improve image quality of quantitative PET/MR techniques in TR&D1.