A knot that bounds a smooth disc in the 4-dimensional ball is called a slice knot. The set of all knots modulo slice knots can be given a group structure under the operation of connected sum. This group is known as the smooth knot concordance group. I will discuss some recent results of T. Cochran, S. Harvey, and myself that show that knots in a certain family whose slice status was previously unknown are in fact not slice.