And many more...

3D Aviation -- L-29 Delfin instruction and flights in Chino, California, USA.
Aeronca L-16 Flight Instruction -- Maryland, USA.
Airborne Aviation -- Tiger Moth and Chipmunk flight training, and scenic / aerobatic / formation flights in Sydney, Australia.
Air Combat Australia -- Fly a MiG-15 in New South Wales, Australia.
Aluminum Overcast -- A historic flight experience in EAA's B-17 Flying Fortress.
American Aeronautical Foundation -- Fly in a B-25 Mitchell or C-47 Skytrain.
Antique Airways -- Tiger Moth, Staggerwing and Beech 18 flights in Queensland, Australia.
Australian Jet Adventures -- Fly a BAC-167 Strikemaster jet in Melbourne, Australia.
Barnstormers Aero Services, Inc. -- Flights and instruction in T-6s, Stearmans and more in Centennial, Colorado, USA
Belle Aire Aviation -- Waco YMF-5 biplane flights in Memphis, Tennessee, USA.
Bi-Plane Adventures, Inc. -- Fly an N3N-3 biplane in Kennesaw, Georgia, USA (near Atlanta).
Big Sky Adventures -- This Johannesburg, South Africa-based firm lets you chose from an amazing selection of warbirds, from supersonic jets to classic biplanes.
Biplane Adventures, Ltd. -- Fly an SNJ-4 (T-6) in San Diego, California, USA. Also offers biplane sightseeing flights and air combat flights.
Biplane Rides -- Fly a Stearman or Waco biplane in long Beach, California, USA.
Classic Wings -- Fly in a deHavilland Tiger Moth biplane and more at Duxford, Cambridgeshire, UK.
Discovery Air Tours -- Fly in a DC-3 in Sydney, Australia.
Dixie Wing of the Commemorative Air Force -- Offers flights in a variety of vintage aircraft.
EAA Fantasy Flight Camp -- Spend two days learning about the history of the B-17 before actually flying in one in Oshkosh, Wisconsin, USA.
Fantasy of Flight Museum -- Various aircraft flights in Polk City, Florida, USA.
Fighter Command -- T-6 Texan flights in Front Royal, Virginia, USA.
Fighter Pilot Pty Ltd. -- Fly in a Strikemaster or Jet Provost near Melbourne, Australia. Mock combat flights available.
Fightertown Aviation's "Skywarriors" Program -- Fly in an L-29 or Iskra jet in Denison, Texas, USA.
Fly an English Electric Lightning or Hawker Hunter in Capetown, South Africa.
Fly Jets -- This Russian company can arrange flights in several types of aircraft from an airfield near Moscow.
Fly a MiG-25 to the edge of space -- Take a Foxbat up for a Mach 2.5 spin.
FlyMagic Flight Experiences -- Unusual and truly unique flight experiences in the UK, including warbirds, aerobatic flights, etc.
FlyMiG -- Fly legendary fighters over Moscow!
Flying Fighters -- Take an adventure flight in any one of a large number of different aircraft types, from jets to Tiger Moth biplanes, in Brisbane, Australia.
Flying Tigers -- Flight adventures in a Scottish Aviation SK-61 Bulldog in Bridgeport, Connecticut, USA.
Fouga Flights -- Aerobatic/recovery training and orientation flights in a Fouga Magister in Kenosha, Wisconsin, USA.
Gauntlet Warbirds -- Warbird, aerobatic and tailwheel training center in Aurora, Illinois, USA, that offers flights in an L-29 Delfin jet and an SNJ/T-6 Texan.
Geelong Flight Centre -- Fly a DH-82 Tiger Moth in Geelong, Australia.
Great New Zealand Biplane Adventure Co. -- Biplane rides in Taupo, New Zealand.
Gulf Coast Wing of the CAF -- Fly a B-17, "Texas Raiders" in Houston, Texas, USA.
Incredible Adventures -- Fly MiGs, T-33s, Lightnings, etc., plus many, many other unique opportunities.
International Fighter Flight Centre (IFFC) -- Strikemaster, Jet Provost and CT-4 Air Trainer flights in Australia.
"J's Bird" -- Offers Harvard familiarization flights in Georgia and other USA locations.
Jet Alpine Fighter -- Fly in a B-25 Mitchell bomber over the Swiss Alps!
Jet Provost Club -- Flights and instruction in the Hunting Jet Provost in Doncaster, UK.
Jet Warbird Training Center -- Familiarization flights and flight training in New Mexico, USA.
Lima Lima -- T-34 flight demo group which also gives rides in Illinois, USA.
Mach2MiG -- MiG-21 flights in Illinois, USA.
MiGFlyer.com -- Fly Russian jets, from the L-39 to the MiG-29, In Russia.
Moscow Aircraft Group -- Fly all kinds of Russian jets in Russia.
Mustang Flights -- Mustang and Winjeel flights at Caboolture Warplane Museum (near Brisbane), Australia.
North American Top-Gun -- Fly T-6 Texans in air-to-air combat.
Olde Thyme Aviation, Inc. -- Old biplane rides in Seattle, Washington, USA.
Preston Aviation -- J-3 Cub and PT-17 Stearman instruction and sightseeing flights at Peach State Airport, Williamson, Georgia, USA.
Pride Aircraft -- L-39 Albatros jet familiarization flights and instruction in Illinois, USA.
Sky Fighters Aerial Combat -- Fly T-34s in Colorado, USA.
Sky Warriors -- Fly T-34s in Georgia, USA.
Stallion-51 -- Dual-control TF-51 Mustang familiarization flights and full LOA (type rating) checkouts.
Texan Flight -- Fly an authentic SNJ-5 Texan in Northern Virginia and Washington, D.C. areas.  
Texas Air Aces -- Fly T-34s in mock combat.
Thunder City -- Mike Beachy Head's jet operation in South Africa. Fly Lightnings, Buccaneers, Hunters, and more.
Tiger Club -- Vintage, aerobatic and formation flying in the UK.
Tom Reilly Vintage Aircraft -- Hands-on restoration training and B-25 Mitchell flights in Florida, USA.
Top Gun Flights -- L-39, Yak-52 and other flights available in Sydney, Australia.
Transport Command, Ltd. -- T-6 rides at Shoreham Airport, West Sussex, UK.
Vampire Preservation Society -- Provides deHavilland Vampire rides in the UK.
Vintage Aircraft Company -- Stearman and SNJ-4 flights in Sonoma, California, USA.
VintageFlight -- DC-3 flights and flight training in Corona, California, USA.
Vintage Flights -- Sightseeing and aerobatic flights over Kennesaw (Atlanta), Georgia, USA in Stearmans and AT-6 Texans. Instruction available.
Voentour M -- Flying adventures in Russia, including flights in MiG-21s, MiG-23s, MiG-25s, Su-27s, etc.
Waldo Wright's Flying Service -- Stearman (and New Standard) biplane flights in various USA locations.
Warbird Adventures, Inc. -- T-6 flights and instruction, Kissimmee, Florida, USA.
Warbird Airventures -- T-6 flights in Nashville, Tennessee, USA and other locations around the USA.
Warbird Aviation -- T-28D Trojan rides and air combat flights in Queensland, Australia.
Warbird Aviation Rendezvous, Inc. -- SNJ-5 (T-6) flights in Bloomington, Illinois, USA.
Warbird Experiences -- T-6G Texan/Harvard rides in Kent, UK.
Warbird Flights, Inc. -- T-6 Texan flights, training and tailwheel sign-offs in Titusville, Florida, USA (Part of the Valiant Air Command Warbird Museum.)
Warbird Heritage Foundation -- Stearman and T-6 rides in Waukegan, Illinois, USA.
Warbird-Rides.com -- Beechcraft RC-45 (Twin Beech) flights and transition training in Stockton, California, USA.
War-Bird Squadron -- Mustang, Spitfire, and Bf-109 combat flights in the USA, UK and Spain.
Warbirds Unlimited Foundation -- B-25H Mitchell rides in the USA. (TBM Avenger flights.)  
"Wings of Freedom" Flight Experiences -- B-24, B-25 and B-17 bomber flights with the Collings Foundation in the USA.
Yakity Yaks -- Nanchang CJ-6 flights and training in Oregon, USA.

Museums that offer flights:

American Airpower Museum, Farmingdale, New York, USA. Historic C-47 flights.
Arizona Wing, Commemorative Air Force, Mesa, Arizona, USA. Flights available in an SNJ, C-45, B-17, and O-2.
Canadian Warplane Heritage Museum, Hamilton, Ontario, Canada. Offers Stearman and Harvard flights.
Cavanaugh Flight Museum, Addison, Texas, USA
Collings Foundation, Stowe, Massachusetts, and Houston, Texas, USA. Offers flights in B-17, B-24, P-51, A-4, and F-4.
Dixie Wing, Commemorative Air Force, Atlanta, Georgia, USA. Flights available in a P-51 Mustang, SBD Dauntless, RAF Expeditor VIP Transport and a PT-26 Trainer.
EAA AirVenture Museum, Fantasy Flight Camps, Oshkosh, Wisconsin, USA. Flights available in a B-17 Flying Fortress and other aircraft.
Flying Tigers Warbird Restoration Museum, Kissimmee, Florida, USA.
Genesee Warbirds, Batavia, New York, USA. Offers experience flights in classic jet warbirds.
Historic Aircraft Restoration Museum, St. Louis, Missouri, USA.
Indiana Aviation Museum, Valparaiso, Indiana, USA. Offers flights in a P-51 Mustang, A-37 Dragonfly, F4U Corsair, and T-28 Trojan.
Lone Star Flight Museum, Galveston, Texas, USA. Flight experiences are available in a B-25 Mitchell, B-17 Flying Fortress, and Boeing PT-17 Stearman.
Mid-Atlantic Air Museum, Reading, Pennsylvania, USA. Historic flights available in a PT-17 or SNJ.
Planes of Fame, Chino, California, and Valle, Arizona, USA.
Tennessee Warbirds Flying Museum, Memphis, Tennessee, USA.
Warbirds West Air Museum, El Cajon, California, USA. Flights available in a T-34, T-28, C-45 and L-29 Delfin jet.
Wings of Eagles Discovery Center, Horseheads, New York, USA. Offers 30-minute flights in a T-6/SNJ, Stearman, and PT-19.
Yankee Air Museum, Belleville, Michigan, USA. Flights available in a B-17 Flying Fortress and B-25 Mitchell bomber.