The industrial applications of Mg, the lightest structural metal, and abundant in Earth's crust, are hampered by its low intrinsic ductility and low fracture toughness at room temperature which is attributed to the underlying less symmetric and plastically anisotropic hcp crystal structure. The thermally activated pyramidal-to-basal (PB) transition of edge/mixed part of the pyramidal dislocations, from the easy-glide pyramidal planes into immobile basal-dissociated dislocations, is largely responsible for the poor Mg ductility. The addition of small amounts of rare-earth elements (RE = Y, Gd, Ce, Nd, etc.) has been shown experimentally to improve ductility by a substantial amount in resulting solid solution Mg alloys, along with the activation of the crucial pyramidal dislocations that were absent in pure Mg. The research work presented in this thesis aims to uncover, via conducting molecular dynamics (MD) simulations, the atomic-scale dislocation mechanisms responsible for the enhanced activation of the pyramidal dislocations in certain ductile Mg alloys, and devise a strategy for designing new ductile Mg alloys. We start by testing the hypothesis that RE solutes might stabilize the edge/mixed pyramidal dislocations on the easy-glide pyramidal planes by increasing the energy barrier of the detrimental PB transition. The finite-temperature MD simulations in Mg-Y alloys, employing a newly developed MEAM potential, establish that Y solute atoms are unable to alter the energy barrier and transition time of the PB transition. We, next, propose a new mechanistic theory to explain the increased activity of the pyramidal dislocations in ductile solid solution Mg alloys. Cross-slip and double cross-slip of screw parts of the pyramidal dislocation loops are capable of acting as a natural dislocation source, and thus a much faster cross-slip process can extenuate the dislocation immobilization effects of the relatively slower PB transition. The energy difference between the low-energy pyramidal II and the high-energy pyramidal I dislocations in pure Mg leads to a slow cross-slip process which is ineffective in defeating the PB transition. We show that small addition of certain favorable solute elements can reduce the pyramidal dislocations energy difference and accelerate the cross-slip process to the levels much faster than the PB transition, and thus enable the enhanced slip of the pyramidal dislocations with a concomitant increase in ductility. We conduct NEB simulation in the Mg-Y alloy to test some aspects of the theory. The evidence for the solute enhanced cross-slip process is presented from the transmission electron microscopy (TEM) observations in Mg-Y alloys. We, furthermore, develop a quantitative model to establish the conditions for ductility as a function of alloys compositions. We then apply the developed quantitative model to the binary, ternary, quaternary, and higher-order dilute solid solution Mg alloys. The predictions of the model are found to be in an excellent agreement with the experimentally observed ductility in a wide range of Mg alloys. The theory, in particular, identifies Mn, Li, Sn, K, Ca, Sr, RE, and Zr solutes as effective in improving the ductility in resulting alloys. The model further predicts an upper limit of concentrations of strongly favorable solutes, such as REs, beyond which ductility begins to deteriorate in Mg alloys.