The most diffucult problem to overcome pertaning to space travel is the enormous amounts of fuel and mass neccessary to make today's space propulsion systems work. Just to lift the relatively small mass of a communications sattelite (and especially the larger masses such as the Space Shuttle) into low Earth orbit, enormous amounts of fuel and and extremely expensive pieces of equipment are needed, resulting in a cost of $1,000s per pound of payload sent into space.
Due to the law of action and reaction, the only way to accelerate an object in one direction is by having that object expel mass in the opposite direction. So the faster and farther you want to travel, the more mass you need to bring with you. This can quickly become a problem when you want to travel between planets or stars, or even into Earth orbit, at a reasonable expense.
These problems call for a new form of propulsion that can produce thrust effeciently, and without expelling mass. Several such devices have been conceived that may possibly attain these amazing goals, but have not yet been conclusively proven successful. This site contains some information on, and links to information on, the attemps of scientists, inventors, and amature tinkerers to perfect these devices.