Investigations are under way to try to find out how beefburgers on sale in UK and Irish Republic supermarkets became contaminated with horsemeat.
Irish food safety officials, who carried out tests two months ago, said the products had been stocked by a number of chains including Tesco and Iceland stores in the UK.
They said there was no human health risk and the burgers had been removed.
Tesco said it was "working... to ensure it does not happen again".
The UK's Department for Environment, Food and Rural Affairs (Defra) said it was working with the Food Standards Agency (FSA) to "urgently investigate" how the products came to contain horsemeat.
The investigation will trace the meat back to its source to "find the cause of the contamination".
After a meeting with food industry representatives, the FSA said it would continue its review of the traceability of the food products concerned.
The FSA also said it would consult local councils and authorities in Ireland over whether legal action should be taken against any of the companies involved.
The Food Safety Authority of Ireland (FSAI), which is conducting similar inquiries, said the meat had come from two processing plants in the Irish Republic - Liffey Meats and Silvercrest Foods - and the Dalepak Hambleton plant in North Yorkshire.
The burgers had been on sale in Tesco and Iceland in the UK and the Republic of Ireland, where they were also on sale in Dunnes Stores, Lidl and Aldi.
A total of 27 burger products were analysed, with 10 of them containing traces of horse DNA and 23 containing pig DNA.
In addition, 31 beef meal products, including cottage pie, beef curry pie and lasagne, were analysed, of which 21 tested positive for pig DNA.
All have been withdrawn from sale.
Horsemeat accounted for approximately 29% of the meat content in one sample from Tesco, which had three frozen beefburger products sold in both the UK and Ireland contaminated with horse DNA.
FSAI director of consumer protection Raymond Ellard said: "The companies have taken a very responsible attitude. On a voluntary basis they have withdrawn products from sale, so have the retailers.
"A long chain of inquiry has to take place now to look at all the raw ingredients that we use for these productions, where they came from and how the cross-contamination could have occurred."
Tesco group technical director Tim Smith stressed the company "immediately withdrew from sale all products from the supplier in question" after receiving the test results on Tuesday.
Mr Smith said food safety and quality was "of the highest importance to Tesco" and "the presence of illegal meat in our products is extremely serious".
He added Tesco was "working with the authorities in Ireland and the UK, and with the supplier concerned, to urgently understand how this has happened and how to ensure it does not happen again".
FSAI chief executive Prof Alan Reilly said there was "a plausible explanation for the presence of pig DNA in these products, due to the fact that meat from different animals is processed in the same meat plants".
But he added: "There is no clear explanation at this time for the presence of horse DNA in products emanating from meat plants that do not use horsemeat in their production process.
"In Ireland, it is not in our culture to eat horsemeat and, therefore, we do not expect to find it in a burger.
"Likewise, for some religious groups, or people who abstain from eating pig meat, the presence of traces of pig DNA is unacceptable."
'Quality and integrity'
Irish Agriculture Minister Simon Coveney reassured the public that the burgers posed no health risk, adding that the Republic of Ireland "probably has the best traceability and food safety in the world".
Iceland said the FSAI's findings were concerning, stressing the company had "withdrawn from sale the two Iceland brand quarter pounder burger lines implicated in the study".
It said it "would be working closely with its suppliers to investigate this issue and to ensure that all Iceland brand products meet the high standards of quality and integrity that we specify and which our customers are entitled to expect".
Aldi said only one of its products - the Oakhurst Beef Burgers (8 pack), which was on sale only in the Republic of Ireland - had been affected.
In a statement, Aldi Stores (Ireland) said it had "immediately removed the product from sale and have launched an investigation into the matter".
The company said it "takes the quality of all its products extremely seriously and demands the highest standards from its suppliers".
Lidl was not available for comment when contacted by the BBC.
Meanwhile, Silvercrest Foods and Dalepak both said they had never bought or traded in horse product and that they had launched an investigation into two continental European third-party suppliers.
A spokesman for Dalepak said: "Analysis is taking place right now but we won't have more information until that analysis is complete."
Conservative MP Anne McIntosh, who chairs the Environment Select Committee, said the news was "stunning... considering all the care that is taken in this country, and all the efforts with food labelling".
She added: "Labelling of produce works very well when you can trust the source of the food and I think we need to be just as careful with our frozen food as with are with our fresh foods."