Some do, some don't is the short answer. As standard, most agreements do not cover maintenance, but in the same way as you can add optional extras to the specification of your car, you can do so to your lease agreement, albeit for a higher monthly fee.
Still, that extra cost brings peace of mind and makes budgeting easier, so make sure you check with the dealer so you know whether or not maintenance is included.
Maintenance packages can include some or all of the following: servicing, repairs, new tyres and batteries, exhausts, breakdown recovery and MOT testing.
While all new cars are covered by a manufacturer's warranty for the first three years, servicing costs and wear and tear items such as tyres are not covered by this.
Cars will not need an MOT test until they are three years old. There's more about this in our Wear and Tear Guide.
Fixed costs: no nasty surprises when you need four new tyres, or the service is due.
Peace of mind: keeping your car in perfect fettle and having the associated servicing paperwork up to date reduces the risk of additional fees when you hand it back.
Check the small print because different leasing companies will offer different components; for example, if you'll need a courtesy car while yours is serviced, make sure it's included.