As Gordon McDougall, chairman at the 144-year-old Arlington, in Glasgow, says, "We have a great social mix of people here.
"We have a smattering of well-known people, who appreciate the privacy. We have some footballers and retired captains of industry rubbing shoulders with prominent trade unionists, church leaders and former police officers.
"As you can imagine, it makes for some interesting conversations in the steam room and the sauna."
The Arlington can't, of course, name any names, but one celebrity who was once here was Rudolf Nureyev, right, the world-renowned ballet dancer.
He came here many years ago for a swim and a massage while he was in performance at the King's Theatre.
The baths club, which lies just off Woodlands Road, has, like its 'friendly rival', the Western Baths Club, been celebrating the awarding of 'Category A' listing status.
The recognition has been conferred by Historic Scotland and means that the striking old building is now of national or international significance.
The Arlington was founded in 1870 and opened in August of the following year. Though many of its original fittings remain - notably the 21m-long pool, above which are suspended a trapeze and travelling rings, and the fantastic-looking Turkish suite with its beehive-shaped roof - it has moved with the times.
Its facilities include a steam room and saunas, slipper baths and hot tubs, a state-of-the-art gymnasium, a weights room, a rather nice cool room, and more.
Its 1000 members can book yoga and exercise classes, or the services of a personal trainer. There's a stylish members' lounge - and, as befits this connected age, free wifi.
"It's fantastic news, especially as we are the world's oldest baths club owned and run by members," says Gordon of Historic Scotland's decision. "For us, it's a recognition of the work that these Victorian pioneers did in creating the Arlington."
General manager Andrew McGilp, who has been here for just 10 months, already finds that the Arlington has got under his skin, adds: "It's also a recognition of an iconic piece of the history of Glasgow that has served Woodlands, and the wider area, for 144 years."
The building was originally designed by the then-fashionable architect John Burnet. The Turkish room ('a tribute to the Alhambra', says Gordon) followed in 1875.
Two other extensions were added before the club had marked its first 40 years.
"So it opened first as a pool then developed a games room, a reading room and a snooker room, and it became a social facility as well as a health facility," says Gordon.
"They liked to mix their work, rest and play."
Club treasurer Tim Pearson said: "Part of the reason they expanded into the social side was that every bathing club has to pay its bills.
"If you get enough members in, you can pay the bills, so by building the facilities, they could increase the numbers of people coming through the doors."
The Arlington remains a non-profit organisation, which means, as Gordon says, that "every penny that is raised in here is spent in here.
"For me, that's very much a community thing. Not only are you sustaining a historical local resource, but you are also putting something back.
"It's a balance between having enough members to pay the bills and not being too busy.
"There are some places that can be overcrowded, but the last thing you want here is not being able to use the gym or the pool, or not being able to get into the Turkish room or the sauna."
The Arlington's archives are held a short walk away in the Mitchell Library, and Gordon is intrigued by the social history they contain.
"You could trace your family history in them if you wanted to," he says. "There are all the membership forms that date all the way back to 1870.
On a rough estimate you could say that some 50,000 people have been members here.
"If your great-grandpa, or your great-great granny was a member, and you're now living in, say, Florida, you could come back here and see their actual signatures on the forms.
"You can wander around here and, armed with her membership number, discover the hook where your great-great granny would have hung her woollen bathing costume.
"And of course you could swim in the same pool that she swam in. There's a real connection with history here, you can feel it the moment you dive in."
It may well be that the Arlington marks its 150th anniversary in six years' time with some sort of publication reflecting all that amazing history.
The baths club can also claim to have given the world the sport of water-polo. Its very first baths-master is said to have developed the sport in this very pool, having first conceived of it in Aberdeenshire.
The Arlington is about to embark on an ambitious exercise with the well-known architects, Page\Park. Its expert team will assess the entire building, from the subsoil to the top of the brick chimney (the tallest in Woodlands), to restore and replace any historic features, to examine how under-used spaces can be pressed into service, and increase the social space in the Arlington's wet areas.
The old place may have its roots in the 19th century but it is determined to make sure it's fit for the 21st.