Throughout history there were many theories on how we could meet or exceed the speed of light, many were extremely fanciful but for the sake of understanding them I shall attempt to explain them here.
There are five common theories: RELATIVITY, FIELD, FORCE, TUNNEL and WAVE.
1. RELATIVITY: As a theory this one was the easiest to deal with, to prove or disprove. It worked on the idea that Einstein's Theory of Relativity was to be literally understood. That to the stationary observer breaking the light barrier was impossible as all forms of measurement would indicate that a craft approaching light was either increasing in mass to the point where it could no longer accelerate or would appear to become energy. The paradox would be if the vessel, on slowing from light speed, would then appear to convert back to matter.
The strength of this theory was primarily based on the concept of the Big Bang for indeed if the "3minute Universe" did exist then we know that at least at one point in history the speed of light was far exceeded by what must have been a finite amount of energy. The theory was proven incomplete for as the craft approached three quarters of the speed of light it's mass relative to the universe did appear to increase even on board. There was no actual friction to slow the craft down but instead it's own gravity effect replaced drag. Top speed was .78 light beyond which we could proceed no farther.
2. FIELD: The field theory is one that has been entertained by many for centuries. The idea is that huge powerful generators could create a field of null-space around the craft reducing the vehicle's relative mass to zero or near zero and then the vehicle could be propelled by simple thrust through space at unimaginable speeds. On analysis with even the most sophisticated of artificial intelligence systems and yes, even a few RAEB systems, the physics for making this actually work are phenomenal. At no point have we been able to create even a workable mathematical structure without having at least one variable represented by infinity. The energy required to generate a cohesive bubble of energy around a craft is hard enough but then to have one that would warp space creating a pocket of null-space within the bubble is untenable at best with existing technology if it is indeed even possible.
3. FORCE: Using a combination of technologies FORCE travel works like this. A vehicle with a magnetic ram scoop activated ahead of it begins a rapid pace toward a large gravitational source, a sun for example. The magnetic ram scoop collects and injects high amounts of stellar material funneling that material into its central driver, concentrating the material there while building up speed around the sun. With the magnetic containment housing nearly full and the craft whipping around the sun at increasing speeds the craft then begins a slingshot breakaway in the direction it wishes to go WHILE opening the tiniest of holes in the aft part of the magnetic containment housing so that the collected material explodes out behind the craft. The theory that this would propel the craft into the faster than light realm (on the assumption that there was one) with a velocity that would eventually dissipate was fraught with the following concern: Mathematically if the craft was travelling fast enough to tear a hole in the fabric of space-time and thereby rupture past normal physical laws it would most likely form a black hole which could effectively pull the gravity base used for acceleration (in this case the sun) in after it. Although it would obviously be easier to travel if you brought your gravity base with you wherever you went the damage to the system you left and the place you arrived at would be catastrophic to say the least, assuming you and the universe survived the trip.
TUNNEL: Most similar to the fictional "STAR GATES" or "Wormholes" the biggest problem with this concept mathematically is energy. No less than five points in the equation require variables equal to infinity to make this concept work in theory and then you're left with the infrastructure problem. No one wants a wormhole or anything else that can completely contort the fabric of space and time that close to them, the cancer rates would nothing compared to the gravitational shock-waves generated.
WAVE: The WAVE theory has also been called SLIPSTREAMING or PHASE Travel because of what it accomplishes. Similar to the FIELD theory in that a pocket of null-space is required.