One of the hallmarks of diffusion NMR and MRI is its ability to employ restricted diffusion to probe compartments much smaller than the excited volume or the MRI voxel, respectively, and extract microstructural information from them. Indeed, the single-pulsed-field-gradient (s-PFG) MR methodologies are employed with great success to probe microstructures in various disciplines ranging from chemistry to neuroscience. However, s-PFG MR also suffers from inherent shortcomings, especially when specimens are characterized by orientation or size distributions. In such cases, the microstructural information available from s-PFG experiments is limited or lost. The double-PFG (d-PFG) MR methodology, an extension of s-PFG MR, has attracted attention owing to recent theoretical studies predicting that it can overcome some inherent limitations of s-PFG MR. In this review, we survey the microstructural features that can be obtained from conventional s-PFG methods in the different q-regimes, as well as the limitations of these methodologies. The experimental aspects of the d-PFG methodology are then presented, along with an overview of its theoretical underpinnings and a general framework for relating the MR signal decay and material microstructure. We will describe the new microstructural features that can be obtained using the d-PFG MR framework. We then discuss recent studies that validated the new theoretical framework using phantoms in which the ground-truth is well known a priori, a crucial step prior to application of d-PFG in neuronal tissue. The experimental findings are in excellent agreement with the theoretical predictions and reveal, inter alia, zero-crossings of the signal decay, enhanced sensitivity towards size distributions, and angular dependencies of the signal decay from which accurate microstructural parameters such as compartment size, and even shape can be extracted. Finally, we show some initial findings in d-PFG imaging. This review lays the foundation for future studies, in which accurate and novel microstructural information could be extracted from complex biological specimens, eventually leading to new contrasts in MRI.