A slice knot is a knot that bounds a smoothly embedded disc in the 4-ball. The set of all knots modulo slice knots can be given an 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. This result has implications about the structure of the Cochran-Orr-Teichner filtration of the knot concordance group. I will also discuss similar results concerning a family of links.