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Do they have these? I know there's fell running of course, but do they have steep rocky races like say up Cadair Idris etc where it's too steep to actually run and you have to scramble?
Yes, lots of hill races are on terrain that you're not 'running', lots of the Lakes fell races are too steep too. In terms of climbing races (which is what I think you're asking) , it's not a scene I'm aware of.
Like the Glencoe Skyline race?
Scaffolder from work entered the Ring of Steall Skyline a couple of weeks back, he runs 5 miles, and he'd been up Ben Lomond twice.
Predictably, didn't end well.
skyrunning i think is the current term.
Are you looking for the Vertical Kilometre type of thing which has a race series across the world? There's a Mamores one as part of the Glencoe Skyline weekend.
on a similar note, and something i was thinking about the other day - are there mtb hill climb races. ie fastest to the top of snowdon for example. Also, are there downhill running races?
Glencoe skyline looks brilliant but depending on how picky they are they might not have me.
The vertical km one also looks great as does the Ring of Steall but that's a bit far for me!
Well up for this though - I hate road running but this really appeals.
Lots in this part of the world. There's a limit to the what the Gendarmes are prepared to "sécuriser" but the terrain is as "climbing" as you can get without people getting seriously injured - and that has happened.
I did the Winter ski-alpinisme races for 25 years, this one 15 times:

Also, are there downhill running races?
Yes. Carnethy do a downhill race from the top of Caerketton to the Ski Centre bottom car park- 1.8km, 300m descent. There will be others
Downhill running race seems like a recipe for broken bones though.
For very short, the Coniston Gullies race is nuts.
For a bit longer, the Grisedale Horseshoe is incredible - some great pics here: https://fellephant.com/?p=3575
Not sure if it still takes place, but there used to be a 'Tryfan Dash' race. Top of Tryfan, down to the car park layby on the road. I think the record was something like 6-7 minutes!
Downhill running race seems like a recipe for broken bones though.
No worse than the downhill part of a race up and down. Fell racers aren't exactly known for cautious and considered descents.
Vertical kilometre races as mentioned, also most mountain ultras IME have sections where you need to use your hands as well as your feet. Pretty sure the Winter ski-alpinisme races @Edukator mentions are what Killian Jornet does when he's not running, AKA skimo racing.
Try it Molly, it's brilliant, like 2 different races, some folk that I beat to the top will kick my arse descending. Just entered my first race since pre covid, prepared for a bad result, not my usual mid table obscurity.