Damp walls in themselves are bad– that cold, constant airborne moisture is bad enough by itself, causing coughs and infections.
But it doesn’t come alone – it is a breeding ground for bacteria – usually mould, a fungus often in the form of ‘black mould’ which loves to nest in timber, paper, plaster board. You’ll get that musty/mildew smell, sometimes the cause is obvious, sometimes not.
Mildew BTW is just another mould, a little easier to clean, but still a fungus.

The fungus releases toxins which attack lungs, causing coughing and wheezing – which can turn to more severe flu-like symptoms – especially in those prone to respiratory tract infections.

A leaking cistern or a badly fitted shower curtain/door will be obvious.
But if the seal around the bath or shower tray is old you won’t see the effect of it (though you’ll probably smell it). A leaking pipe behind the bath or under the sink can be hard to spot too.
One giveaway is your pump (if you have a pumped system) switching on & off when you’re not using water. That is evidence of either a dripping tap/toilet or a hidden leak.
It won’t go away, so don’t ignore it.

Drying clothes in the bedroom or on a rad, fish tanks breathing and (most often) leaving the ensuite door open after showering. If you do that the warm moisture heads straight across the room to the window where it forms condensation. That in turns rolls down the glass and forms a nice warm breeding ground for fungus.

There is an extractor fan in the bathroom (and the kitchen) for a reason. Leave it on and clean it every couple of months, even with your vacuum cleaner. Close the bathroom door – that makes it easier for the fan to do its work. If you must open the door, you must open a window nearby.
And never block vents. They allow tired (oxygen-less) air to go out and fresh air to come in.

Exterior: Heat wants to escape, that’s a fact. You may have heard of thermal bridging, typically it happens where an inside wall is in direct contact with an exterior surface – think of your inside wall where it meets the window or floor; or where the ceiling meets the roof. All of that lovely expensive heat is just looking for the easiest way out. You can get mould build-up there too – that’s where the moisture from the air settles.
Of course, an exterior leak can also be caused by missing roof tiles, defective guttering or the ‘pointing’ between bricks wearing away. In older properties uninsulated north facing walls can absorb and transmit moisture from out to in.

Treatment: You can buy ready made mould killing solutions or you can mix 3-parts warm water to 1-part bleach (wear gloves). Rub it in well and leave for an hour at least before you start cleaning.

If the mould is well established you need a professional, experienced, cleaner. It can be more difficult to get rid of the black staining that’s left behind than it was to kill off the mould. How To…. 

Repainting: Depending on whether you’ve killed the mould and how bad the staining was, you may want to use a strong sealant coat over the area, or you might want to use an anti mould additive in the paint. Ask the guy in the shop.
And use a soft sheen or a satin emulsion – if the problem does recur, it makes cleaning it off a lot easier

BTW – you may sometimes see peeling ceilings in bathrooms with no sign of mould. New bathroom ceilings should have been sealed to stop the steam getting in behind. But of course that doesn’t always happen. If you see this, scrape the ceiling really well, get a proper (I like PVA but again ask the guy) sealant and apply that before painting.