Morriston Hospital Accident and Emergency Department in Swansea had its highest ever levels of attendance yesterday, with more than 300 patients presenting for care.
Abertawe Bro Morgannwg University Health Board says it has seen a high number of emergency admissions, and its A&E departments are extremely busy and under pressure.
Cardiff and Vale University Health Board has also apologised to patients for delays in treatment, particular in the emergency unit at the University Hospital of Wales.
It says there has been "exceptional demand on our services", and "in some instances we have not met the standards we set ourselves."
Hywel Dda University Health Board has apologised for forcing patients to wait in ambulances outside Bronglais Hospital in Aberystwyth.
Mark Brandreth, Director of Planning and Operations at the health board, said: "Any patient waiting in an ambulance is assessed, managed and treated according to their needs by our doctors and nurses."
Bosses say ambulance admissions across Wales are currently 22 percent higher than predictions for this time of year.
Some planned surgery has been postponed, to provide more resources to the emergency department.
The health board is asking people with minor injuries or illnesses to try other options, such as getting advice from a local pharmacist or GP, or using minor injury units, rather than going to Bronglais.