Hard to be sure from the pix, but it looks like the dirt (lawn) is higher that the driveway. So you could either allow it to slope down the meet the concrete or you could build a curb to hold back the dirt. If you choose to let it slope, then all you really have to do is bevel that sharp edge at the end of the existing curb. probably can do it with a concrete saw. If you want to wrap the curb around, choose your material and get busy. You could pour a concrete curb or lay in Belgian blocks on end. Or build with the dry stack blocks that you can find at the bigbox.