We prove several results concerning the existence of low-discrepancy point sets with respect to an arbitrary non-uniform measure μ on the d-dimensional unit cube. We improve a theorem of Beck, by showing that for any d ≥ 1, N ≥ 1, and any non-negative, normalized Borel measure μ on ${[0,1]}^d$ there exists a point set $x_1,...,x_N\in {[0,1]}^d$ whose star-discrepancy with respect to μ is of order $D_N*(x_1,...,x_N;\mu )\ll \left({\left(logN\right)}^{(3d+1)/2}\right)/N$. For the proof we use a theorem of Banaszczyk concerning the balancing of vectors, which implies an upper bound for the linear discrepancy...