If you have any joint debts with a spouse or any other party who have died, the debt will become your sole responsibility. This is true if you were a guarantor for a debt owed by the deceased.
When someone dies, their debts become a liability on their estate. The executor of the estate, or the administrator if no will has been left, is responsible for paying any outstanding debts from the estate.
When the debt is not in joint names, if there is insufficient money or assets in the estate to pay off all the outstanding debts, then the debts would be paid in priority order until the money or assets run out. Any remaining debts are likely to be written off.
Surviving relatives will not usually be responsible for paying off any outstanding debts for the deceased unless they acted as a guarantor or it was a joint debt.