Do I qualify?

You've found a place you love and can picture all your stuff in it.  Now's time to turn in the application.  What are your chances of being accepted? 
We look at 3 main things:  how's your credit, how's your income, and what does your landlord say about you.  If you have current late payments on your credit report, an eviction, or undischarged bankruptcy many landlords will not accept you. 
We will ask for proof of income and we want your income to be at least 3x the rent.  If you receive paychecks we want paystubs.  If you're self-employed we want tax returns and bank statements.
If your financials are less than stellar, what can you do?  Some landlords will accept co-signers.  Another option you may be offered is an increased deposit if you're a higher risk.  If you have extenuating circumstances like being a victim of identity theft disclose that upfront.
We will also call your most current landlord and possibly the one before to ask if you paid on time, took care of the property and whether they'd rent to you again.
Remember as owners or agents for owners we want to know that you will provide us with reliable income and protect our property. 
If you're having trouble meeting the standards for a property management company try looking for units rented directly by owners.  It may be easier to plead your case face to face with the property owner.