The Welsh Ambulance has declared an "extraordinary incident" across the country.
The service said it is due to "handover delays at hospitals across Wales."
Although nationwide, the incident relates particularly to the Swansea Bay area.
The issue is the length of time ambulance crews are having to wait before handing patients over once they arrive at hospital.
"You can help us by only calling 999 if your emergency is life or limb-threatening. Please call 111 to receive non-urgent medical advice," the service said on X, formerly known as Twitter.
In a statement from the service they say, "We experienced exceptionally high levels of demand with very sick patients over the weekend and we were at full capacity. This resulted in long delays to ambulance handovers which, in turn, affected the availability of ambulances in the community.
"We’d like to thank the Welsh Ambulance Service for their support as well as neighbouring health boards and would like to appeal to the public to only attend our Emergency Department if they have no other option.
"Pressures on our unscheduled care continue, and we are appealing to the friends and families of patients in hospital across Swansea Bay to help us discharge them on time.
"Through reducing unnecessary delays to patient discharges we can improve the flow of patients through the hospital which, in turn, will help speed up ambulance handovers so that they can get back out into the community more quickly.
"It is also better for patients’ wellbeing to go home as soon as they are well enough, as unnecessary prolonged stays in hospital can impact their recovery and affect their wellbeing.
The problem of handover delays has been a persistent issue for the WAS for some time now.
Last year ITV Cymru Wales' Health Reporter, Katie Fenton, spent time with ambulance crews in Swansea, struggling to handover patients.
She learnt how paramedics were spending their entire 12 hour shifts sitting outside hospitals waiting to offload patients.
One of those crew members, Mark Tonkin, told Katie how the job had become beyond difficult.
"It's hard, it's demoralising, it's quite saddening to know that we could potentially be coming in on shift, just sitting round listening to the calls going out on the radio when we should be out there actually getting to the calls."
Earlier this year an emergency consultant at Wales' largest hospital has admitted that patient care is being "compromised" by unprecedented demand and delays in discharging patients as well.
Dr Katja Empson has worked at the University Hospital of Wales' emergency unit for 14 years, and told ITV News she has never seen it this challenged.
"Things are certainly worse than I've known them, the experience that people are having through the system, the delays getting to hospital and then the delays getting back out of hospital, they are really worse than they used to be," Dr Empson explained.
"There [are] patients sitting in chairs in corners of the department where we really shouldn't be asking people to wait.
"When every single cubicle and nook and cranny of the department has got a patient sitting in it, your capacity to manage all of those patients safely, with compassion and kindness becomes challenging, particularly towards the end of a long clinical shift."
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