He may but it depends.

• If the damp is down to something which needs repairing – such as a leaky roof, then yes, this will fall under the landlords repairing obligations.
• If there is nothing that is actually in disrepair but the property generally is in a poor condition, such that it could be considered to be ‘unfait for human habitation’ then your landlord may be liable under the fitness for habitation rules.  Note though that this legislation is aimed at very serious issues so a small bit of damp in one corner of the ceiling will not qualify.
• However often damp in properties is caused by ‘lifestyle issues’.  So if you never open any of the windows, keep the heating low and dry all your clothes indoors, this will create damp which may lead to mould.  

Condensation, damp and mould are often very serious problems in rented property but it is often difficult to work out whether they are down to the condition of the property or the lifestyle of the tenant.

If you don’t think it is you, a good start would be to contact your local authority and ask them to come out and do an inspection.  They should be able to let you know what the issue is and if they think it is serious, they may order your landlord to do repair or other works.

See also the article below.