Aside from warp technology, it would appear that the transporter is the least likely to happen right now, however, the dream still lives on and the possibilities are enormous! Scientists have taken theories posed by Einstein, a little bit of inspiration from Star Trek, and a lot of work and innovation working toward the goal of transporter technology. We have made great strides in this area and we are currently able to transport data, but not matter.
In a way we have been doing this for decades with fax machines. You scan the information, encode it, send it over a phone line, and reproduce it on the other end. According to a 2007 article by Steve Connor scientists have successfully “teleported data over a distance of 89 miles” utilizing quantum entanglement (I won’t even pretend that I understand what these people are talking about. I’m a Librarian, not a physicist), which does not require an intermediary source such as a phone line to transport the information. To the untrained bystander the data seems to appear out of thin air. In fact, the teleporters of the Star Trek franchize were actually modeled using the theory of quantum entanglement when they were thought of four and a half decades ago.
The hope for future research in this field is to test the theories in space. The study that was highlighted by Connor in his article and was perfomed by Dr. Rupert Ursin, et al. was published in Nature and was used as a feasibility study performed by the European Space Agency.
As recent as 2009, Ursin and his team have been conducting research and tests in order to determine the viability of communicating in this manner.
In the Star Trek universe, the first transporter that was created by humans, and was able to tranport matter was first developed by Emory Erikson prior to 2121. That’s only a little more than a century off. With the rapid rate of technological advancement we have seen over the past one hundred years, I’d say it is possible to breach the inability to tranport matter by that time.