Moving to new rental accommodation - deposit/rent questions

I'm looking to move out of my current accommodation and plan to give my notice shortly. However, I am wondering how it works if I find a suitable place before my notice period is up. Would a landlord normally hold a place for a couple of weeks, or would I find myself in a situation where I am paying rent for my current accommodation and my new place during that time? How does it usually work?

By the way, I meant to add that I do not have a lease agreement with my current landlord, and I pay the rent monthly

You give one month's notice to the old landlord
Normally rent is paid in advance so u have to pay for the month in question
After that you can leave when you like
The only way you can avoid paying rent for the overlap period is to ensure the commencement of the new tenancy coincides with the end of the old tenancy. To do that requires a bit of good luck and planning.. the new landlord might well agree to a delay in the starting date of a lease due to the slump in demand for rental accomodation at the moment .. just ask and see what happens.. by the way if you have paid a deposit initially on the old place you should expect to get it back unless you are in breach of the tenancy agreement (normally its one month rent as a deposit)

A landlord would need his head examined if he let a place go to a 'less desirable' tenant for the sake of 2 weeks rent. Most will hold the place for at least a week if it means getting a perceived better tenant. Some landlords may be under such severe financial pressure that even a few days lost rent is too much and that's fair enough but most should be able to accomodate if you seem a decent prospect.