They use both. Weapons incorporating fusion (boosted fission bombs, or "true" Teller-Ulam fusion weapons) have several advantages. Fusion fuel is cheaper than weapons grade plutonium, per unit of yield. Fusion weapons scale up to higher yields. Fusion weapons deliver more yield in a compact package. That's why the world's nuclear arsenals are overwhelmingly thermonuclear, with few pure-fission weapons.
On that note, here's an interesting podcast episode about nuclear weapons that explains their history and current configuration for us laymen: http://omegataupodcast.net/270-nuclear-weapons/
