In the US, debt is generally tied to an entity called your estate (your financial entity post death, basically). Debt - for example let's say you have a big credit card debt - will be paid by the assets of your estate. Different types of debt often have different priorities (when it comes time for the executor or court to decide what gets paid first or last). As your estate is settled, the debts usually get taken care of from the assets (basically a simple subtraction matter). If there aren't enough assets in the estate to pay the debts, then the creditor is almost always out of luck.

Just to be clear, this system is essentially identical to "debt is inherited, but never in an amount greater than the value of the inherited assets".