The tax-efficient five-year term accounts were first launched in January 1991, offering savers tax-free interest on total investments of up to £9,000.
The accounts are subject to a maximum investment in year one of £3,000, and up to £1,800 in the next four years, subject to the £9,000 overall limit.
With the original Tessas due to mature in January next year, there were fears that banks and building societies would see millions of pounds being withdrawn as investors took their rewards.
However, the Chancellor, Kenneth Clarke, announced measures in his Budget last November to encourage investors to keep their money in the accounts. People whose Tessas mature will now be allowed to take the interest earned tax-free and reinvest the entire capital saved over the five years, up to the £9,000 limit, in a new Tessa for a further five-year term.
Savers have six months from the maturity date of their Tessa to decide to reinvest the funds. And if interest rates continue to creep up, many will find the savings scheme an ideal shelter for their cash and help Tessas to regain the popularity that saw one million people open accounts in the first three months of their existence.
Most of the main banks and building societies offer Tessas, with the best rate currently being offered by Sun Banking, formerly Confederation Bank.
The account requires an up-front investment of £8,900 to be deposited in a feeder account with Sun Banking. It will transfer the maximum amounts allowed each year from the account into a Tessa, paying a fixed gross rate of 9 per cent per annum.
This turns the first year's Tessa investment of £3,000 into £3,270 after 12 months. The investor has the choice of taking the interest tax-free or leaving it in the account, where the interest that accrues on the interest will also be available free of tax.
The cash remaining in the feeder account waiting to be transferred into the Tessa in future years also attracts interest at 9 per cent, although this is subject to tax.
The original £8,900 deposit is enough to make sure the maximum Tessa limit is taken advantage of, because interest on the cash in the feeder account will eventually bring the sum available to invest above £9,000.
The 9 per cent fixed rate is guaranteed to turn the investor's £8,900 into £13,243 at the end of the five-year term, an increase of 48.8 per cent.
Everyone over the age of 18 is allowed to hold one Tessa account, and they are particularly attractive for the higher-rate tax payer who would normally pay 40 per cent tax on any capital gain.
Norman Brown, 75, of Littlehampton in West Sussex, invested £7,500 in a Tessa with Sun Banking, then Confederation Bank, in 1991 at a fixed rate of 7.5 per cent.
The retired solicitor is now looking forward to realising the benefits of the scheme, which he says will have seen his capital increase to around £12,100 when the account matures next January.
And the father of four has seen enough of a Tessa's tax advantages to convince him to reinvest the proceeds in a new scheme. He said: "I have a range of equity-based investments, and I like to balance these with government stocks and National Savings Certificates.
"However, I think Tessas are a wonderful tax-free investment, particularly for the person who pays tax at the higher rate.