On Australia Day 1966, the three Beaumont children, Jane, 9, Arnna, 7, and Grant, 4, left their parents' home in Adelaide for a day at Glenelg beach.

They were among hundreds, if not thousands, of people who flocked to the coast to seek relief from the stifling heat.

Yet, despite being surrounded by so many, so few seemed to notice them and none could say definitively where they went and who they were with.

The case has been the subject of an intense police investigation, media scrutiny and public conjecture for more than half a century.

Suspects have been questioned and ruled out, a number of extensive searches have been conducted and detectives have travelled interstate and overseas in search of a breakthrough.

But no trace of the children has ever been found.

South Australia Police insist the case remains open and the investigation is still active.

Only last year, officers converged on a factory site in Adelaide's west that was once owned by a key suspect, Harry Phipps, after new computer technology uncovered unexplained anomalies in the ground behind the buildings.

For several years Phipps, who died in 2004, has been considered a person of interest and police have been unable to rule him out or say for certain that he was involved.

He bore a resemblance to an identikit picture prepared at the time of the disappearances and had lived close to Glenelg Beach.

Phipps also had a reputation for handing out crisp one-pound notes, and a woman who worked across the road from the beach told police the children had come into her shop and bought food, paying with a one-pound note.

More recently two men came forward to tell a story of how they dug a large hole at the North Plympton factory site at the request of Phipps on the Australia Day weekend.

An excavation in 2013 found nothing, though it was apparently done in the wrong spot.

A new excavation in 2018, uncovered only animal remains.

In the end, it was like so many of the theories and clues that have come to nothing.

One early report said the children were living in the Mud Islands, in Victoria's Port Phillip Bay, which were duly, and unsuccessfully, searched.

In 1968, the entire crew of the British freighter Devon was questioned and photographed in New Zealand.

And then several years later a Perth woman came forward to claim that for about nine months in 1966 she had lived next door to the Beaumont children in an isolated railway town near the SA-WA border.

But it was the visit of Dutch clairvoyant Gerard Croiset that probably generated the most interest.

He claimed to have had a compelling vision that the children had been buried less than two kilometres from where they went missing.

Mr Croiset believed the children were buried under a brick kiln in a nearby warehouse.

The extensive excavation of the areas he pointed to were carried out, without result.

In 1997, a former investigator on the Beaumont case, Stan Swaine, said he believed a woman living in Canberra was actually Jane Beaumont.

Police checked his claims and questioned the woman concerned but were convinced she was not the eldest of the three missing children.

Several years ago there were suggestions the children were alive and living in New Zealand.

Then, just before the 40th anniversary of their disappearance, Tasmanian Police Commissioner Richard McCreadie said he believed convicted killer James O'Neill could easily have kidnapped the three children.

McCreadie admitted he had no evidence to support his theory and SA police said they had interviewed O'Neill and ruled him out.

Less than a month later detectives travelled to Victoria where they were thought to have interviewed child killer Derek Percy who had spent more than 30 years in Ararat Prison for the rape, torture and murder of a 12-year-old girl.

Percy was said to have travelled to Adelaide around the time the Beaumonts went missing, but he was also ruled out.

Police believe a breakthrough in the Beaumont case would be "significant for the country".

For the children's parents Nancy and Jim Beaumont it would mean justice or at least finally knowing what happened to their children.

Sadly, for Nancy, who passed away in Adelaide this week, this is no longer possible.

Australian Associated Press