When I was renting, my landlord was fine with me painting the walls. He supplied the paint, and I did the work. If I chose to paint walls in weird colors or did a mural, we agreed that I would either paint over that with a neutral color if I ever moved out, or he would be required to charge me the cost incurred for his painters to repaint it.
Now I have a friend who lives in an apartment, and her landlord does not allow for that. If you want the apartment painted and it's been at least two years, they will send someone to paint it for you. That's because not everyone is careful and many in the past have just painted over outlets, light switches, etc without removing the face plate. Or they get paint all over the hardwood floors.
It really depends on your landlord and any rules laid out in your lease. In many cases it falls under the category of "making alterations to the property" and not allowed without permission. Does not hurt to ask them, however. That way you are clear about it.