Dynamical characteristics of flexible polymer molecules in nanoscopic confinements are primarily dictated by the relative values of the confinement length scales with respect to the polymer persistence length. Depending on whether the channel height is larger [1] or smaller [2] than the polymer persistence length, altogether different polymer dynamics is ensued, as illustrated in the pioneering theoretical studies by de Gennes [1] and Odijk [2]. Rapid advances of nanofabrication and polymer handling, over the last few years, have been able to provide experimental validation to these studies and at the same time have been able to unravel different intriguing physical issues unique to nanoconfinement induced dynamics of polymer molecules. These studies have led to a plethora of new applications ranging from the estimation of structural and mechanical properties of polymer to fabrication of novel, portable diagnostic tool kits. In this review article, we shall revisit different physical and technological issues involved in polymer dynamics in nanoconfinements. First, we shall identify the effect of varying degrees of confinement (de Gennes and Odijk regime [1,2]) on the stretching dynamics and the overall representation of the polymer molecule. Next, we shall discuss the possible physical interaction forces on the polymer molecule introduced by the presence of the confining walls and the resulting effects like formation of wall adjacent depletion layers, asymmetric distribution of polymer mass density from the wall to the channel center line etc. Thirdly, we will highlight the possible effects of a background field (flow field or electric field or a combination of both and they may bear signatures of the involved nanoscopic length scales) on the overall nanoscale polymer dynamics. We will also briefly discuss the technological intricacies involved in the relevant nanofabrication and polymer handling schemes and also the issues governing the polymer dynamics modeling involving such scales. Finally, we shall conclude by indicating the possible directions, anticipated outcomes, and significances of future research in polymer transport and dynamics in nanoscopic confinements.