Our spacecraft use a propulsion system developed by the Nazis and no one has been able to create anything different in eighty-six years.
Flown non-chemical rocket propulsion or attitude control systems:
Ion drives:
Hall Effect Thrusters
Field-Emission Electric Propulsion
Gridded Ion Thruster
Colloid Thruster
Electrodeless Plasma Thruster
Pulsed Plasma Thruster
Resistojet
Arcjet
Solar Sail
Reaction Wheel
Lightcraft (for very small values of 'flown')
Developmental Systems*:
Nano-Particle Field Extraction Thruster
VASIMR
Helicon Double Layer Thruster
Microwave Electrothermal Thruster
Magnetoplasma Dynamic Thruster/Lorentz Force Accelerator
Pulsed Inductive Thruster/FARAD
Radio-Isotope Thruster
Vacuum Arc Thruster
MagBeam
Nuclear Pulse (Orion/Daedalus/Longshot/Medusa)
Antimatter Catalyzed Nuclear Pulse
Magneto-Inertial Fusion Rocket
Pulsed Fission Fusion Propulsion
Mass-Driver
Fission-Fragment Rocket
Diffractive Solar Sail
Fission Sail
Fusion Rocket
Nuclear Thermal Rocket
Electric Sail
Magnetic Sail
MagSail
Mini-Magnetospheric Plasma Propulsion (M2P2)
Magnetoplasma Sail
Plasma Magnet
Plasma Magnetoshell
Beam Powered Magsail
Electrodynamic Tether
Momentum Exchange Tether
Universal Orbital Support System
Alcubierre Drive
Casimir Effect
Differential Sail
Laser Augmented Sail
Laser Electric Propulsion
Laser Thermal Rocket
Mass Beam
Microwave Sail
Microwave Thermal Propulsion
(Yes, I got carried away down a wiki wormhole)
* Ranging from mission ready to doable with current technology, to doable with massive infrastructure, to the physics says this should work but we don't have a clue how to implement it (the Alcubierre Drive).