I bought a car in 2007 and dealership took my previous car as a trade in. Next month I went back and pd down payment as agreed to dealership. After that month my account was transferred to a finance company which I have been making monthly payments to. Now 3 years later after no notifications by phone or mail, dealership is stating that I owe them still for down payment. My question is, is there a statute of limitations for owing the car dealership?

Statute of Limitations refers to the amount of time someone has to initiate legal action against someone. SOLs vary so there is no way to answer your question without knowing your general locale. ANY question about law needs to include your area because laws vary.

Whether the SOL has passed or not doesn't mean they can't dun you for the money.

Did you pay by check? Money order, cash? Did you get a receipt?