This paper is concerned with the almost sure exponential stability of the multidimensional nonlinear stochastic differential delay equation (SDDE) with variable delays of the form dx(t) = f(x(t - delta(1)(t)), t) dt broken vertical bar g(x(t - delta(2)(t)), t) dB(t), where delta(1), delta(2) : R+ -> [0, tau] stand for variable delays. We show that if the corresponding (nondelay) stochastic differential equation (SDE) dy(t) = f(y(t), t) dt + g(y(t), t) dB(t) admits a Lyapunov function (which in particular implies the almost sure exponential stability of the SDE) then there exists a positive number tau* such that the SDDE is also almost sure exponentially stable as long as the delay is bounded by tau*. We provide an implicit lower bound for tau* which can be computed numerically. Moreover, our new theory enables us to design stochastic delay feedback controls in order to stabilize unstable differential equations.