In general, a car loan isn't assumable by another party unless your financing agreement specifically allows it. Loans are granted to an individual based on his own credit profile, and changing the borrower can increase or decrease the lender risk. Anyone looking to take over the payments for you generally will have to apply for financing and satisfy the lender that he'll be able to fulfill the obligation.
Check Your Paperwork
Some lenders allow you to transfer the loan or lease agreement to another party -- as long as that individual meets the financing qualification. The person hoping to assume the car loan will have to fill out an application, document his income and debt obligations, and have his credit score examined. If the application is approved, the loan and possession of the car is transferred to the new buyer, who subsequently takes over the payments. If the buyer's credit isn't as strong as yours, you may be able to modify the loan, with the new buyer taking over the balance, but incurring a higher interest rate.
Another option for transferring the loan is to refinance in someone else's name. You'll need to call your current lender to get the payoff amount for your loan, and have the new borrower apply for that amount of financing. If it's with a different lender, you'll have to provide the information on your car so the lender can determine the vehicle's value and whether it's worth the money being borrowed. Once the refinance is complete, the loan and ownership transfers.
If you're looking for someone else to assume your car lease, you'll generally find it an easier task than for a traditional loan. However it's up to you to find a replacement and negotiate the terms. Websites such as SwapALease.com and LeaseTrader.com help eager sellers find qualified buyers. Check your paperwork -- some lenders restrict the ability to transfer leases at certain points of the contract. For example, as of this publication, Nissan won't allow a lease transfer if there are fewer than seven payments remaining, and BMV won't do so if there are fewer than six. There also are fees involved, charged by both the leasing company and any third-party site that facilitates the transaction. Finally, be prepared to accept something below what the lease payments currently are, and even offer a cash incentive to the new owner to stimulate interest. Losing money on the transaction may be less of a hit than sticking with an obligation you can't afford or paying the surrender fees to turn the car in early.