Attenuation correction is an essential component of the long chain of data correction techniques required to achieve the full potential of quantitative positron emission tomography (PET) imaging. The development of combined PET/magnetic resonance imaging (MRI) systems mandated the widespread interest in developing novel strategies for deriving accurate attenuation maps with the aim to improve the quantitative accuracy of these emerging hybrid imaging systems. The attenuation map in PET/MRI should ideally be derived from anatomical MR images; however, MRI intensities reflect proton density and relaxation time properties of biological tissues rather than their electron density and photon attenuation properties. Therefore, in contrast to PET/computed tomography, there is a lack of standardized global mapping between the intensities of MRI signal and linear attenuation coefficients at 511 keV. Moreover, in standard MRI sequences, bones and lung tissues do not produce measurable signals owing to their low proton density and short transverse relaxation times. MR images are also inevitably subject to artifacts that degrade their quality, thus compromising their applicability for the task of attenuation correction in PET/MRI. MRI-guided attenuation correction strategies can be classified in three broad categories: (i) segmentation-based approaches, (ii) atlas-registration and machine learning methods, and (iii) emission/transmission-based approaches. This paper summarizes past and current state-of-the-art developments and latest advances in PET/MRI attenuation correction. The advantages and drawbacks of each approach for addressing the challenges of MR-based attenuation correction are comprehensively described. The opportunities brought by both MRI and PET imaging modalities for deriving accurate attenuation maps and improving PET quantification will be elaborated. Future prospects and potential clinical applications of these techniques and their integration in commercial systems will also be discussed.