There are many other methods that are used for transfecting cells in culture. These include:
- Cationic lipid mediated (as mentioned by user137)
- Calcium Phosphate mediated transfection
- Biolistic methods
Newer techniques include optical transfection (using lasers to bore a hole through membrane) and impalefection (using nanotubes). Wikipedia page on transfection nicely summarizes these techniques.
Induced pluripotent stem cells can be cultured from a patient, transformed using the plasmid and transplanted back to the tissue where it can replace the old cells with "corrected" cells.
There is another somewhat recent technique called in-vivo electroporation using which Nucleic Acid can be delivered directly in to the desired tissue.
However, to fix a mutation or introduce a transgene, just transfection is not enough; you have to make sure that the desired gene is inserted in the chromosome at the right location. Faulty gene can be replaced by the transfected functional gene via homologous recombination(HR), but rates of recombination are generally very low. Targeted gene insertion methods using genome editing tools such as Zinc-Finger Nuclease (ZFN), TALEN, and Crispr-Cas can increase this rate by introducing a break in the DNA at that region; this causes DNA repair systems to "repair" the break using HR.
There is another type of DNA repair system called NHEJ (Non-Homologous End Joining), and this is not what we are interested in. NHEJ can, in fact cause mutation through indels. Presence of the transfected DNA would shift the preference towards HR, however it has also been shown that different stages of cell cycle have different preference towards HR and NHEJ (cant find the reference as of now but I am sure about it because I was thinking at some point of time, to work on this).
Having said that, I would add that it is essential to screen the cells to see if the gene is incorporated or not. If some optimal protocol is devised then one can proceed to in-vivo electroporation.