Traveling involves basically three physical variables: speed, distance, and time. To get somewhere faster, you must either increase your speed, shorten the distance to the destination, or somehow slow down local time for the traveler. Let's take a look at how each of these methods have been utilized by writers in the past.
GOING FASTER
This is the method of choice for two of the biggest franchises in sci-fi - Star Wars and Star Trek. Star Trek alone made the term warp speed a pop culture substitute for describing anything going really fast. But what exactly is it? Roddenberry's literary team had a real knack even back in the 60's for coming up with fictional names for technologies that sounded almost plausible. Just the term "warp" brings to mind relativistic concepts of bent space or time. I always assumed that they were simply fractions of light speed, but that made the stars which would fly off the edge of the view screen seem ridiculous. Apparently the intent was to represent multiples of the speed of light. This Wikipedia entry states that the very first pilot episode of TOS, The Cage, refers to the warp drive as producing a time warp, which gets around the Einstein limit but not the method used to accomplish it. Later episodes link warp speed to subspace, which is also used to explain how communication messages can get back to Starfleet faster than light travels. But subspace is just a made up term with no analog in the world of science. Of course, the opposite of subspace would be the equally made up hyperspace, which is the method of explanation chosen in the Star Wars franchise, enabled by the so-called hyperdrive.
SHRINKING THE DISTANCE
The only way to find a path shorter than a straight line between two points in space is to just bend the space, which is theoretically possible thanks to General Relativity. The only viable candidate in existing theory to accomplish this is an Einstein-Rosen bridge. It is a kind of tunnel through space-time, commonly referred to as a wormhole. We see this method used for example at the space stations Babylon 5 and DS9. What always made me chuckle is that an E-R bridge lives at the center of a black hole, where ordinary matter would be crushed under the enormous gravitational forces. Also, there's no way to control where you might end up on the other side, and even if you could enter it, the journey would be almost instantaneous, not through some long swirly corridor as is often depicted. I've recently read about more stable types of wormhole theories, but they require an exotic form of negative energy that probably does not even exist. There does seem, however, to be enough wiggle room at the boundaries of physics (see this link) with which you could build a barely plausible wormhole fiction.
SLOWING THE CLOCK
The final way to shorten interstellar travel time is to slow down the aging process of the traveller, and the most plausible method has more to do with biology than physics. I'm talking about the cryogenic freeze, as we encounter aboard the Nostromo in Alien, or the Hunter-Gratzner in Pitch Black. Of course, this doesn't really count as "speedy" interstellar travel because even if the crew doesn't age, the rest of the world outside still does, and it doesn't make the actual travel time any shorter. But at least it is within the realm of future technological advancement. If you take a more physics level approach you can slow down time itself by traveling close to the speed of light, where Special Relativity says you won't age as fast as the world outside. But you don't escape the problem of everyone you know back on Earth dying of old age before you reach your destination. And besides, traveling at that speed requires so much energy you couldn't do it by bringing fuel along (as illustrated in this cool analysis of the Relativistic Rocket).
Think of any science fiction book, film, or series that involves space travel and you'll most likely encounter one of these methods being used. Anything else is bound to start crossing over into pure fantasy.
SHRINKING THE DISTANCE
The only way to find a path shorter than a straight line between two points in space is to just bend the space, which is theoretically possible thanks to General Relativity. The only viable candidate in existing theory to accomplish this is an Einstein-Rosen bridge. It is a kind of tunnel through space-time, commonly referred to as a wormhole. We see this method used for example at the space stations Babylon 5 and DS9. What always made me chuckle is that an E-R bridge lives at the center of a black hole, where ordinary matter would be crushed under the enormous gravitational forces. Also, there's no way to control where you might end up on the other side, and even if you could enter it, the journey would be almost instantaneous, not through some long swirly corridor as is often depicted. I've recently read about more stable types of wormhole theories, but they require an exotic form of negative energy that probably does not even exist. There does seem, however, to be enough wiggle room at the boundaries of physics (see this link) with which you could build a barely plausible wormhole fiction.
SLOWING THE CLOCK
The final way to shorten interstellar travel time is to slow down the aging process of the traveller, and the most plausible method has more to do with biology than physics. I'm talking about the cryogenic freeze, as we encounter aboard the Nostromo in Alien, or the Hunter-Gratzner in Pitch Black. Of course, this doesn't really count as "speedy" interstellar travel because even if the crew doesn't age, the rest of the world outside still does, and it doesn't make the actual travel time any shorter. But at least it is within the realm of future technological advancement. If you take a more physics level approach you can slow down time itself by traveling close to the speed of light, where Special Relativity says you won't age as fast as the world outside. But you don't escape the problem of everyone you know back on Earth dying of old age before you reach your destination. And besides, traveling at that speed requires so much energy you couldn't do it by bringing fuel along (as illustrated in this cool analysis of the Relativistic Rocket).
Think of any science fiction book, film, or series that involves space travel and you'll most likely encounter one of these methods being used. Anything else is bound to start crossing over into pure fantasy.