I tried to grow basil inside last winter, but within a couple days of sprouting, the little pot molded over and the seedlings died. Gross. This last Christmas, I was given a second chance by way of an indoor herb growing kit that included a ton of peat pellets and a little greenhouse to start the seeds off in. I have a huge west-facing bay window in my apartment that gets sun most of the day, and below the bay window is a baseboard heater that provides additional warmth. Needless to say, they are doing much better this year!
They are: oregano, basil, dill, chives, and thyme. Hopefully they will be ripe for the picking (or at least ready to transfer outside) in a couple months! The basil already smells like basil!