What should I use to fill gaps between floorboards?

Save the sawdust from the final sanding and mix it with a 2 part epoxy resin.
This will give a finish that's the same colour as the original boards.
You might need to fill the larger gaps with mastic first if your floor is on joists to stop the epoxy falling through.
Be sure to leave space to apply 5mm of epoxy on top of the mastic so it's flush with the boards.

Big Softy
February 2009