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Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line Mohamed H Amsaad Doctor Of Philosophy in Mathematics It is known from Fluid Mechanics [25], [39], [57], [62], [66], [75], [83], [84], [90], [91], [93], [105], [108], that the time-evolution of a probability measure describing some physical quantity (such as the density of a fluid) is related to the velocity of the fluid by the continuity equation. This is known as the Eulerian description of fluid flow. Dually, the Lagrangian description uses the flow of the velocity field to look at the individual trajectories of particles. In the case of flows on the real line, only recently has it been discovered [70], [92] that ``some sort'' of dual Lagrangian flow consisting of monotone maps is always available to match an Eulerian flow. The uniqueness of this ``monotone flow'' among all possible ``flows'' (quotation marks used precisely because traditionally it cannot be called a ``flow'' unless it is unique) of the fluid velocity which is the centerpiece of this dissertation. The Lagrangian description of absolutely continuous curves of probability measures on the real line is analyzed in this thesis. Whereas each such curve admits a Lagrangian description as a well-defined flow of its velocity field, further conditions on the curve and/or its velocity are necessary for uniqueness. We identify two seemingly unrelated such conditions that ensure that the only flow map associated to the curve consists of a time-independent rearrangement of the generalized inverses of the cumulative distribution functions of the measures on the curve. At the same time, our methods of proof yield uniqueness within a certain class for the curve associated to a given velocity; that is, they provide uniqueness for the solution of the continuity equation within a certain class of curves. Our proposed approach is based on the connection between the flow equation and one-dimensional Optimal Transport. This is based on joint work of the author with A. Tudorascu [21], in which some results on wellposedness (in the one-dimensional case) have been achieved. The results are presented in major conferences and published in a highly ranked mathematical journal.