Your tenants are entitled to serve you with a Notice to Quit and must do so if they wish to vacate the property. This can happen with periodic tenants or tenants in a fixed term who wish to end the tenancy early. If your tenants leave during the period of their tenancy agreement, you can take court action against them to be reimbursed for unpaid rent.
If you paid your deposit on or after 1 April 2013, your landlord must protect it in an approved tenancy deposit scheme. These schemes work a little differently for joint tenants. Even though you might have paid your portion of the deposit directly to the agent or landlord, the landlord will protect the full deposit for the tenancy as one payment. This normally means that only one tenant, the lead tenant, will have the power to start the dispute process if you disagree with how the landlord has returned the deposit at the end of the tenancy.