A lease assignment, often called a lease takeover or a lease transfer, is the legal term for when your landlord allows you to pass responsibility for your apartment to another tenant. The new tenant, your assignee, becomes the tenant under the lease agreement instead of you. They pay the rent directly to the landlord and are treated as the tenant in every other way.
The catch with lease assignments is that you aren't always completely released from liability under the lease, so if the assignee defaults then the landlord might come after you for that rent. If the lease is joint and several (which most residential leases are) then they could also come after you for the rent if one of the other tenants on the lease were to default.
In general you will need the consent of everyone involved in the assignment in order to pull off a lease assignment: the landlord, you and your assignee. Your master lease or the assignment agreement that your landlord suggests might also require the approval of any cotenants, so look out for that.
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