Face coverings are now mandatory in shops, shopping centres and other indoor settings.
The new regulation comes as confirmed cases of Covid-19 have increased substantially over the weekend, mainly driven by large outbreaks in counties Kildare, Laois and Offaly.
The regulations provide that a person must wear a face covering in such premises, unless they have a reasonable excuse.
The premises involved include shopping centres, libraries, cinemas, and museums, as well as businesses such as nail bars, hair dressers, dry cleaners, bookmakers, tattooists and travel agents.
The regulations do not apply to children under 13 or to workers who are separated from others by a screen or are able to keep two metres away from others.
Those who do not comply could face fines of up to €2,500 or six months in jail, although the Minister for Justice has said garda intervention would be a last resort.
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A GP in Co Laois urged people not to push their masks under their chins, as this will contaminate the mask.
Dr Sumi Dunne said people should fully remove the mask if they want to communicate (from a safe distance) or to eat, rather than push the mask onto their necks.
She said a face covering should cover the nose and under the chin and be secured with ties or ear loops.
Speaking on RTÉ's Morning Ireland, Dr Dunne said a cloth covering should ideally have two layers and it is important that people wash their hands before and after putting one on.
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Face coverings mandatory in Northern Ireland
Face coverings are now also mandatory in certain indoor settings including shops and shopping centres in Northern Ireland.
Staff in shops, children under 13 and those with an illness or impairment are exempt from the legislation.
The Northern Ireland Executive's public information campaign is urging people to 'Wear one for Everyone'.
In a joint statement, the Executive said: "We want face coverings in shops and other enclosed settings to increasingly become the norm in our society. But it also has to be remembered that some people cannot wear face coverings, due to a disability or health condition.
"It is expected that businesses will encourage and promote compliance to help protect customers and staff. We are grateful to retailers for their constructive work on promoting the benefits of face coverings.
"We are confident most people will comply, but those who do not wear a face covering without a reasonable excuse risk being issued with a fixed penalty notice by the police."
People who breach the legislation without having a reasonable excuse could attract a fixed penalty notice of £60 (reduced to £30 if paid within 14 days).
The amount of a fixed penalty notice increases if there are further offences.
The Executive said regulations are kept under review and can be updated if required.
It is expected that businesses will help encourage compliance to help protect customers and staff.
However, the Executive said that responsibility for imposing a fixed penalty notice for non-compliance will rest with the PSNI.
Additional reporting Gail Conway