In general, no. Credit card debts and the like are forgiven, but it is necessary to write a letter to the creditor and enclose an official (noterized) copy of the death certificate. Funeral Homes generally provide the principle relative (wife, child) with 10 copies.

Here are the exceptions:

Co-signer. If someone co-signs a promissory note along with the deceased person, then he/she will be liable even if the co-signature was an accommodation only and did not benefit the co-signer financially.

Guarantee. If someone guarantees the payment of the debt then on the death of the primary obligor, if his estate has insufficient assets to pay the debt, the creditor can demand that the guarantor pay the debt.

Necessaries. If someone has supplied the deceased person with necessities of life, such as food, clothing, medicine, shelter and so on, that also benefit the survivor, then although the primary responsibility is that of the deceased person's estate, the supplier may look for payment to the survivor who benefited from the necessaries.

A will just decides who gets assets after the creditors. The estate is responsible for all debts. If the estate cannot afford the debts, then they are written off.

However, creditors (even unsecured creditors) have priority over beneficiaries.

Any secured debts should be paid out of the estate of the deceased, if there is one.