Let X be a smooth, projective, d-dimensional subvariety of ℙn(ℂ). Barth's theorem says that Hq(X, ΩpX)=0 when p≠q and q+p≤2 d-n (if p=0 we must have q>0). It is very interesting to look for analogous vanishing theorems for Hq(X, ΩpX(m)), m ∈ ℤ (see [S-S], [F], [S]). In this paper we prove some vanishing theorems for Hq(X, ΩpX(1)), for Hq(X, ΩpX(m)) when m≤-1, and, if dim(X)=n-2, for Hq(X, Ω2X(m)) and Hq(X, SkΩ1X(m)). We use standard techniques and some of our previous results.