Let S be a set of n points in R3. Let ω* = ω*(S) be the width (i.e., thickness) of a minimum-width infinite cylindrical shell (the region between two co-axial cylinders) containing S. We first present an O(n5)-time algorithm for computing ω*, which as far as we know is the first nontrivial algorithm for this problem. We then present an O(n2+δ)-time algorithm, for any δ>0, that computes a cylindrical shell of width at most 26(1+1/n4/9)ω* containing S.