Slice alternating links and rational homology 4-balls

I will discuss an ongoing project to try to determine which alternating knots are smoothly slice and the related question of which double branched covers of alternating links bound smooth 4-manifolds with the rational homology of the 4-ball. The starting point is Lisca's classification of slice two-bridge knots using and obstruction coming from Donaldson's diagonalisation theorem.