Sainsbury's and Asda have both said they will stop shoppers buying more than three of any particular food item. Sainsbury's has also said it will prioritise vulnerable and elderly people for online deliveries. Its boss Mike Coupe said the firm "is trying to make sure everyone has access to the items they need".
Asda's buying restrictions will also apply to cleaning and toiletry products. In a blog post the chain said it had seen "a heightened demand" for products both in stores and online.
"We have plenty of products to go around, but we have a responsibility to do the right thing for our communities to help our customers look after their loved ones in a time of need," it added
Aldi has already introduced limits of four items per shopper on all products, while Morrisons has said it will expand its online delivery service. Other retailers including Tesco and Boots have set limits on particularly popular products such as pasta, tissues and hand sanitiser. Boots chief executive Sebastian James said the issue was not supply, but demand.
"No supply chain can survive a sudden, unexpected global ten-fold increase in demand. And what we thought was incredibly important was that as many people as possible could get what they actually needed," he told the BBC's Today programme.
Supermarkets' online delivery services have also been overwhelmed by the surge in demand.
Shoppers at the online retailer Ocado have been venting their frustration on Twitter at being one of thousands in a virtual queue to place a food order.