Date Availability: Does the availability coincide with your moving schedule and current lease?
Pet and guest rule: If the landlord does not allow a pet or a roommate who is not on the lease, an additional charge may be levied and the owner may confront you about the additional guest (whether it be a person or a pet).
Condition: What's the general physical condition of the apartment? If it needs to be repaired, find out what work will be done by the landlord before you move in. Generally, most landlords will have a vacant apartment cleaned and repainted. You may be able to negotiate a smaller security deposit if you have the time and skill to undertake those improvements. But be sure to know your limitations: if you have to repair crumbling plaster, you may end up spending more money than you anticipated. Those renters that are handy or are willing to paint the apartment themselves should bring up that point when negotiating lease terms. Oftentimes landlords will be thrilled to have tenants that are willing to paint & repair the apartment to their own liking!
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