A number is called badly approximable if there is a positive constant c such that |x-p/q| > c/q^2 holds for all rationals p/q, so that close approximation by rationals requires relatively large denominators. The set of such numbers is Lebesgue-null but has full Hausdorff dimension. This set can be viewed as the union over c of the set BA(c) of numbers which satisfy the above inequality for the fixed constant c. J. Kurzweil obtained dimension bounds on BA(c), which were later improved by D. Hensley. We will discuss recent work, joint with D. Kleinbock, in which we use homogeneous dynamics to produce dimension bounds for a higher-dimensional analog.