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Motivated from the risk measures, superhedging in fnance and uncertainties instatistics, the G-Brownian motion {Bt: t≥0}, which is concisely speaking is acontinuous process having stationary and independent increments under a givensublinear expectation E[.], was introduced by S. Peng. The theory of G-Brownianmotion is really still in its infancy. In this dissertation the existence and stability ofsolutions for stochastic diferential equations under G-Brownian motion （G-SDEs）are discussed.The method of upper and lower solutions for G-SDEs is introduced. The exis-tence theory for the solutions of G-SDEs with discontinuous coefcients is estab-lished. It is determined that the G-SDEs have more than one solutions if the driftcoefcients or the second coefcients are discontinuous functions. The solutions ofG-SDEs exist even if frst and second coefcients are discontinuous functions si-multaneously. Comparison theorems are given with the help of upper and lowersolutions.The upper and lower solutions method is applied to the backward stochastic dif-ferential equations under G-Brownian motion （G-BSDEs）. The existence of solutionsis shown for G-BSDEs, whose coefcients may be discontinuous functions. Difer-ent cases are studied considering the frst then second and then both coefcients asdiscontinuous functions. Some examples are examined by taking the Heaviside andsawtooth functions as the coefcients of G-BSDEs. The existence and uniqueness ofsolutions for G-SDEs and G-BSDEs in a generalized space are also studied.The Lyapunov function method is established for G-SDEs. The p-moment ex- ponential stability for G-SDEs is entrenched. It is validated that under the usualconditions the equilibrium positions of the G-SDEs are p-moment exponentially sta-ble. The q-moment instability of G-SDEs is evaluated. Some conditions are ensuredunder which the trivial solutions of G-SDEs are q-moment unstable.