Jack Nicklaus once said that if he had to teach someone to putt then he would start them off by putting with the left hand low. When we putt 'conventionally' it can be a struggle to keep the face square at impact - even with the best putters on the market. The hands can get too active for a variety of reasons, and some of us are using putters that might be too long or that aren't fitted.
People who aren't fans of the left-hand-low method often say they struggle on longer putts but love it on the shorter ones, so why not mix it up? There's nothing to say you have to use the same grip for every putt. Just look at Collin Morikawa.
There are now many ways to hold the flat stick, such as the claw grip or the reverse overlap, but here are some of the benefits of using the left-hand-low method.
A huge benefit is that it gets your shoulders more level. It is easy to have your shoulders too open at address and this will lead to a slight out to in path - this method squares up the shoulders and keeps the putter moving in the right direction and straight down the target, making it easier to hole out more efficiently. The back of the left hand is going straight down the line and a good check is that you should still be able to see some of the face still.
This is pretty straightforward and nowhere near as fiddly as you might imagine if you've never tried it before. Place your right hand at the top of the grip (for right-handed golfers) and then simply place your left hand below it.
This is about getting as comfortable as possible and feeling what is best for you but a great, solid way of gripping the club is to have one finger overlapping. Your lower thumb can then sit straight down the grip. If you still feel like there is too much movement then try double overlapping to give more of a connection.
Another key component is to have the ball a bit forward of centre with the shaft leaning slightly towards the target - you want the left shoulder to drive the stroke and this will help to encourage that. The right hand can very easily get too dominant and with this grip, and your shoulders more level, you will hopefully see some more putts dropping.