Quantitative imaging (QI) is demonstrating strong promise across multiple clinical applications. For clinical translation of QI methods, objective evaluation on clinically relevant tasks is essential. To address this need, multiple evaluation strategies are being developed. In this paper, based on previous literature, we outline four emerging frameworks to perform evaluation studies of QI methods. We first discuss the use of virtual imaging trials (VITs) to evaluate QI methods. Next, we outline a no-gold-standard evaluation framework to clinically evaluate QI methods without ground truth. Third, a framework to evaluate QI methods for joint detection and quantification tasks is outlined. Finally, we outline a framework to evaluate QI methods that output multi-dimensional parameters, such as radiomic features. We review these frameworks, discussing their utilities and limitations. Further, we examine future research areas in evaluation of QI methods. Given the recent advancements in PET, including long axial field-of-view scanners and the development of artificial-intelligence algorithms, we present these frameworks in the context of PET.