  We introduce strong p-completeness and use them for studying the continuous
dependence of solutions of SDE's on non-compact manifolds. We obtain conditions
for the existence of global smooth solution flow, and prove their
diffeomorphism properties. The criterion is in terms of the growth of the
solutions of the linearized equations (i.e. the derivative flows). It is also
given in terms of the driving vector fields and their derivatives. For the
Euclidean spaces there are three principle classes of examples: (1) the driving
vector fields grow at most $|x|^{1-2\alpha}$ and the derivatives of the driving
vector are allowed to be the order $|x|^\alpha$; (2) the coefficients of the
SDE grows linearly while its derivative allowed to grow logarithmically, (3)
The coefficients are bounded while the derivatives are allowed to grow
linearly. We also given test on non-explosions.
