Typically, your lease dictates not just the last date you may occupy the property, but also how many days before your lease is up that 1) you must notify your landlord if you are moving out, and that 2) the landlord may put a keybox on the premises to facilitate showing it to prospective new tenants.
There is no single answer to 1 and 2, but usually these are 30 days or the end of the month, meaning you usually must notify the landlord of your intention to move out by the end of the previous month and he gets 30 days to show it to others while you're still in it.
Once you have notified the landlord, you need to find a new place to live. So, many tenants are looking 30-45 days before their lease ends. If at all possible, you want to move out a couple of days early for 2 reasons. First, it takes time to move and to clean both the new place before you move in and the old place after you move out. Second, electric and other utilities are not turned on instantly -- you have to go to the city or your electric provider and apply -- usually 2-3 business days is how long it takes.
So, the overlap is best to occur at the end of your current lease. For example, February 28 is a Saturday. You need to order utility turn-on for February 27 and be present at the new property. This means for electric service that you ordered the new service on February 24 or earlier. Pack your stuff, arrange for moving van and help as needed, change subscriptions to the new place, order cancellation of utilities, all prior to the moving day. Do not expect any utility to come out on a weekend or holiday to turn on electricity, water or gas.
If your lease expires February 28, your new lease should allow move-in on February 27. You can go back on the 28th and finishing cleaning up and turn the keys over, normally by 5 pm. Yes, it is possible to move in a single day, but it is a real hassle. Any problems with vehicles or other things could run you over the old lease. This is a bad thing. Usually the penalty per day for 'holding over' in the old place is much, much higher than the proration for moving into the new place early. It's best not to take chances, and just start the new lease a couple of days before the 28 and move with less stress.
After you turn your keys in, ask for them to inspect. Nothing requires them to inspect at your convenience, so ask nicely. They have 30 days to give you an accounting for your security deposit and charges. Don't expect them to run right over when you give them the keys, inspect and give you cash. Ain't gonna happen. This often creates problems for tenants who don't have enough security deposit and rent for the new place when they haven't gotten the old security deposit back. Sorry. This is life.
Some places will allow you time to build up your security deposit, say half up front and half after you get the old deposit back, but they don't have to. Some places require 60 days notice to move out. This can also create a gap, because you have to know where you're going but since most places know for sure about vacancies only 30 days out, there will be a risky period of 30 days when you probably don't know where your new place is. If you're on month-to-month, 60 day notice would not apply; and you can ask when you sign the lease to have the 60 days reduced to 30. They might.