In this paper, we are concerned with the energy decay rates for the viscoelastic wave equation with nonlinear damping and nonlinear time-varying delay in the boundary and acoustic boundary conditions. Here we consider with minimal condition on the relaxation function g, namely g ′ ( t ) ≤ − µ ( t ) G ( g ( t ) ) , where G is an increasing and convex function near the origin and µ is a positive nonincreasing function. The decay rates of the energy depend on the functions µ,G and on the function F defined by f 0 which represents the growth at the origin of