In reply to Oujmik:
A corner like this is the classic place for moud to grow on the inside as there is a greater external surface exposed to the elements, whilst the internal surface area is low compared to the volume of wall behind it. It will be almost certain that the source of moisture will be from the outside, so look there first (you've mentioned some possibilities):
1. Check the guttering for any leaks.
2. Check the downpipe that you mentioned - poor joints will let water track down the pipe, across brackets etc.
3. Check the gully into which the downpipe empties. Ensure it's empty and freeflowing. Also make sure the downpipe channels the rainwater into the gully properly and it's not just splashing around and soaking the surrounding ground.
4. You've mentioned plants, - cut them back from the wall, maybe cut them down if they are big enough to be blocking the wind from helping the wall to dry (although if they are that big, they may well be protecting the wall rather than adding moisture). Also check the roots, make sure they're not damging the wall.
5. Look at the ground level: it may have been added to over time, so may need reducing. Also check that it's allowing water to fall away from the house rather than towards.
6. If you have plants outside, is all the ground along the wall drainable? Any hard surfaces (paving, tarmac etc) will trap moisture which can then spread into the wall. Rain will also bounce off hard paving and contribute to damp walls
7. What is the wall construction? Dense, low-porosity materials will hold the water once they are wet. Cement renders and similar, often added to try and block damp, actually make things worse.
Can you post a photo of the outside, please?
Go through all the above points and I'll bet you'll improve things (I appreciate point 7 is not a quick fix). Also do what others have suggested internally (keeping condensation down, heating, ventilation etc).
Finally, and most importantly, do not get any so-called 'professional damp expert' in. They will try and blind you with science (sticking a moisture-meter on the wall - you can see it's damp FFS) and will play on your fears to try and charge a fortune for 'remedies' that are dubious to say the least.
The remedy is often fairly straightforward and can easily be done yourself, just try and take a holistic view, don't panic. Let us know how you get on.