Virtual knot theory concerns knots in thickened surfaces, with the equivalence relation generated by Reidemeister moves and (de)stabilisation of the underlying surface. Concordance is an equivalence relation that corresponds to genus zero cobordisms. This talk is based on joint work with Hans Boden and Micah Chrisman. It introduces various approaches to virtual knot cobordisms. Their multiplicity becomes a strength of the theory and is used to compute the sliceness and the 4-genus for virtual knots with fewer than 6 crossings.