If the buyer is paying cash, an appraisal is not required.
If there is a new mortgage being made to purchase the property, and that mortgage will be delivered to an Agency Investor (i.e., FNMA, FHA, etc.), the lender is required to obtain an appraisal of the property.
The only exception is if the Agency Investor's loan program, provides a Property Inspection Waiver (i.e., PIW, basically an appraisal waiver) which removes the requirement for an appraisal.
