Here we consider a sequence of positive numbers β = {β k } k∈Z n with β 0 = 1, and assume that there exists 0 < r ≤ 1 such that for each i = 1, 2,. .. , n and k = (k 1 ,. .. , k n) ∈ Z n , we have r ≤ β k β k+ϵ i ≤ 1 if k i ≥ 0, and r ≤ β k+ϵ i β k ≤ 1 if k i < 0. For such a weight sequence β, we define the weighted sequence space L 2 (T n , β) to be the set of all f (z) = k∈Z n a k z k for which k∈Z n |a k | 2 β 2 k < ∞. Here T is the unit circle in the complex plane, and for n ≥ 1, T n denotes the n-Torus which is the cartesian product of n copies of T. For φ ∈ L ∞ (T n , β), we define the slant weighted Toeplitz operator A φ on L 2 (T n , β) and establish several properties of A φ. We also prove that A φ cannot be hyponormal unless φ ≡ 0.