1) To take over another person's rights and/or obligations. For example, one person might assume another's car lease, residential lease, or debt. 2) In bankruptcy, for the bankruptcy trustee to take over an unexpired lease or executory contract. The bankruptcy trustee has the right to assume or reject these agreements. If the trustee assumes a lease or contract, he or she can either allow the agreement to continue in force or assign that agreement, if the trustee believes doing so could raise money for the debtors creditors.