  This work focuses on the numerical approximations of neutral stochastic delay
differential equations with their drift and diffusion coefficients growing
super-linearly with respect to both delay variables and state variables. Under
generalized monotonicity conditions, we prove that the backward Euler method
not only converges strongly in the mean square sense with order $1/2$, but also
inherit the mean square exponential stability of the original equations. As a
byproduct, we obtain the same results on convergence rate and exponential
stability of the backward Euler method for stochastic delay differential
equations with generalized monotonicity conditions. These theoretical results
are finally supported by several numerical experiments.
