David's point below is well taken - Why not just get a new loan?
You still have to qualify to assume an existing loan - and in most cases pay an Origination Fee.
You also will have to provide a down payment equal to the difference between existing loan balance and sales price.
The short answer is yes you may be able to assume - but it may not be your best choice.
The mortgage: No, it probably can't be taken over. That's called "assuming" a loan. Loans used to be assumable. Almost none are today. There are a few exceptions, but in general you can't assume someone else's mortgage.
However, you can pay the other person's mortgage. It stays in their name, but you pay. (We'll get to that in a moment.)
The deed: Yes. The name on the deed determines ownership. And someone can deed the property to you. It's very simple. That would make you the legal owner.
But now to practicalities. How do you do it, since mortgages can't (generally) be assumed? You do what's called a "subject to." That is the property is deeded to you "subject to" the existing mortgage. To oversimplify a bit, you agree to make the seller's mortgage payments. At some point in the future, you'll go out and get a new mortgage on the property. That will retire the old mortgage, so that the seller is no longer legally responsible for payments. Note: Subject Tos violate the lender's "due on sale" clause. That's the provision in the mortgage that says a property can't be sold without the lender's prior written approval. If the property is sold, the lender can call the mortgage due and payable--that is, it can foreclose on the mortgage. As a practical matter, in today's economy, very few lenders will do that. They really don't care who's paying the mortgage; they're happy if anyone is. Still, it's important to understand that you would be violating the due on sale clause.
So, that's your answer. The loan can't be assumed. The property can be deeded to you. You'd accomplish that by doing a "Subject To," and at some point in the future you'd get your own mortgage on the property. For more information, talk to a real estate attorney.
Hope that helps.