Why the original party often stays on the hook
Assigning a contract transfers its rights, and the new party's assumption is its promise to perform. Neither step, on its own, ends the original party's responsibility to the counterparty. If the new party fails to perform, the counterparty can usually still look to the one who assigned. Ending that exposure generally takes a novation or an express release signed by the counterparty, which is a separate agreement and has to be negotiated. In New York, whether an anti-assignment clause makes a transfer void or only gives the other side a claim for breach depends on how clearly the clause is written.
Leases and licenses
Commercial leases are a common setting for these questions. Landlords often require consent to assign, and many leases treat a change in the tenant's ownership as an assignment. A landlord may also insist that the original tenant or a guarantor remain liable, and a guaranty that survives an assignment can follow someone for the rest of the term. Intellectual property licenses are another, since many patent and copyright licenses cannot be transferred without consent under federal law even when the contract is silent. Read the specific consent, notice, and recapture language before treating a lease or license as freely transferable.
Assumption and assignment in bankruptcy
Bankruptcy changes the rules. A debtor can, with court approval, assume a contract or unexpired lease and assign it to a buyer, often despite an anti-assignment clause and over the counterparty's objection. In exchange, defaults generally have to be cured and the counterparty given adequate assurance that the buyer will perform, and some kinds of contracts remain hard to assign without consent. If you are the counterparty, the cure amount stated in the debtor's notice is often the main point to contest, and objection deadlines can be short. If you are the buyer, you want to know which contracts come with you and what cure costs you are taking on. A first conversation sorts out which of these settings you are in and what consents or filings are needed.