From what launcher components are currently using composite materials, this slide from NASA's Advanced Composite Structures and Materials Technologies for Launch Vehicles presentation (PDF) tells the story pretty well on composites on current launch vehicles (presentation is from March 2011):
Additionally, some suborbital space planes also use composites, like for example the all-carbon-composite body Virgin Galactic SpaceShipTwo and to a various degree likely all of the Virgin Galactic suborbital fleet, including SpaceShipOne and their motherships that they're launched in-air from, White Knight Two and White Knight, respectively. They're all built by a California based Northrop Grumman corporation aerospace company Scaled Composites.
If Virgin Galactic's future orbital launch vehicle LauncherOne will use any of its parts made of composites is however unclear from what little of its proposed design is known to me, but it will be also launched in air from its mothership (Scaled Composites WhiteKnightTwo) that is a composite based body.
NASA is also looking into the use of composites for future space exploration missions (not merely launchers and crew modules but also probes, landers, rovers etc.) and has announced composites research partnership with Bell, GE, Lockheed Martin, Northrop Grumman, Boeing, United Technologies Corporation and Pratt & Whitney in September 2013.
NASA has previously been working on a design for an Orion crew exploration vehicle pressure shell made entirely of composite materials to gain in-house composites experience, testing composite cryogenic fuel tank and can use a lighter composite payload shroud on some of the launch vehicles if needs be, so they clearly have intentions in using composites more in the future.
SpaceX Falcon 9 also uses what looked to me a carbon-fibre reinforced polymer payload fairing to me (judging by images presented in a webcast stream before its latest flight seven), but it is certainly made of composites, as stated on SpaceX page on Falcon 9. Together with NASA, they also built a carbon-composite carrier structure for the first Dragon spacecraft heat shield (Photo Credit: SpaceX/Roger Gilbertson):