1. To think that something is true even though there is no evidence for it.
2. To start being responsible for something.
3. To adopt or to take on something.
4. To take on a particular role or function.
5. To put on a pretense of something, usually in order to hide one's true feelings.
6. To take to oneself formally (the insignia of office or symbol of a vocation); to undertake (an office or duty).
7. To take for granted as the basis of argument or action; to suppose that a thing is.

To assume means "to suppose, to put forward" as with a hypothesis or a possibility: "Let's assume Marina doesn't get a pay raise; could she still afford to buy a new car?"