A circle can be embedded in 3-dimensional space in many different ways. Such an embedding is called a knot. Two knots K_0 and K_1 are said to be concordant if they are the edges of an annulus embedded in the 4-dimensional space R^3 × [0, 1], where K_0 occurs at time t = 0 and K_1 occurs at time t = 1. Assuming everything is smooth, a concordance can be viewed as a kind of movie with K_0 at the beginning and K_1 at the end. Knot concordance can be generalized to knots in 3-dimensional spaces other than R^3. We will discuss one such generalization, due to Turaev, to knots in thickened surfaces Σ ×[0, 1]. Here Σ is an oriented compact surface. This is generalization is equivalent to virtual knot concordance, introduced by Carter-Kamada-Saito and Kauffman. While classical knot concordance is a highly developed subject, much less is known about the virtual knot case. In this talk, we discuss some recent joint work with Boden and Gaudreau that attempts to shed light on this interesting problem. In particular, we show will show how to determine the slice genus of a large number of the 92800 virtual knots having at most 6 crossings.