  The BMO martingale theory is extensively used to study nonlinear
multi-dimensional stochastic equations (SEs) in $\cR^p$ ($p\in [1, \infty)$)
and backward stochastic differential equations (BSDEs) in $\cR^p\times \cH^p$
($p\in (1, \infty)$) and in $\cR^\infty\times \bar{\cH^\infty}^{BMO}$, with the
coefficients being allowed to be unbounded. In particular, the probabilistic
version of Fefferman's inequality plays a crucial role in the development of
our theory, which seems to be new. Several new results are consequently
obtained. The particular multi-dimensional linear case for SDEs and BSDEs are
separately investigated, and the existence and uniqueness of a solution is
connected to the property that the elementary solutions-matrix for the
associated homogeneous SDE satisfies the reverse H\"older inequality for some
suitable exponent $p\ge 1$. Finally, we establish some relations between
Kazamaki's quadratic critical exponent $b(M)$ of a BMO martingale $M$ and the
spectral radius of the solution operator for the $M$-driven SDE, which lead to
a characterization of Kazamaki's quadratic critical exponent of BMO martingales
being infinite.
