Ion thrusters [0], solar sails [1], nuclear pulse propulsion [2] are all things that have real research being done on them. There's more esoteric stuff (theoretical) such as the Alcubierre Drive [3] (Warp drive!) that's more of the realm of sci-fi.

If you can maintain a constant 1g of acceleration, you'd get to ~0.99c in approximately 1 year -- but you need to carry/provide the energy to fuel whatever propulsion system you're using and the amount of fuel needed to maintain that 1g of acceleration increases as you get closer to c.

[0] https://en.wikipedia.org/wiki/Ion_thruster

[1] https://en.wikipedia.org/wiki/Solar_sail

[2] https://en.wikipedia.org/wiki/Nuclear_pulse_propulsion

[3] https://en.wikipedia.org/wiki/Alcubierre_drive