There are a number of debt collection agencies whose job it is to chase down unpaid debts for other companies.
If you receive a phone call or letter asking for money, the onus is on THEM to prove that you DO owe the money and that it's genuine debt. So if the debts aren't yours, don't panic; send a letter to say you're not responsible. National Debtline has a template letter you can use.
For older debts, the law says you can't be pursued in court for civil issues six years after the event (five in Scotland).
So for a personal loan, credit/store card or bank account where there's been no contact for six or more years, a lender usually can't get a court action against you. This is increased to 12 years for mortgages/secured loans – but for most tax debts there is no time limit.
Bear in mind that contact can mean anything, including making a payment, and that restarts the clock. Courts may also extend the time, for example, if lenders have taken reasonable steps to contact you. Some lenders still chase for the money after that period, but only a court can force you to pay.
If it's happening to you, see the National Debtline factsheet and template letter, or get help before speaking to the creditor.