If, say, the clutch went or there was an fault with the engine etc, does the lease company pay? So like when you're renting a property?
Generally, maintenance isn't included with leases. You can usually add it on, but it makes the monthly payment more expensive. Otherwise, it's just like any other way of having a car: if the fault is covered under the car's warranty, you don't pay. If it's wear-and-tear, you do.
It's one of those things where the headline figures can look attractive, but once you've included the leasing agent fee (which to be fair is sometimes nothing) and any extras, it can start to look less good value.
One of my brothers leased a Land Rover a few years back and when it went back the leasing company (not Land Rover, to be fair) clawed back almost two thousand pounds in 'damage'. It wasn't a bill they were expecting and it took them almost a year to pay it off. Of course, this doesn't happen to everyone but it's worth bearing in mind -- if you're not the type to keep the car absolutely pristine at all times and it gets the odd bollard bump leasing probably isn't for you.
The MSE page on car leasing is actually very good, and has lots of helpful information.
I've looked at it a few times and I can never get the numbers to add up. (I also live in centralish London so I never feel 100% confident that I'll be able to hand something back bump- and scrape-free.)
I own and run two older cars but that suits my personal circumstances, and I also do some of the maintenance myself. Of course not everyone wants to do this so it isn't the right route for other people.