In the UK it is a legal requirement for a fire risk assessment to be carried out on almost all premises where people could be working.
This is under the Regulatory Reform (Fire Safety) Order 2005 in England and Wales, the Fire (Scotland) Act 2005 in Scotland and the Fire and Rescue Services (Northern Ireland) Order 2006 in Northern Ireland.
Under these regulations, the "responsible person" is legally obliged to ensure that the fire risk assessment is carried out and to deal with any problems that were highlighted during that assessment (although the relevant phrase is "duty holder" under the Scotland regulations and "appropriate person" under the Northern Ireland regulations).
The definition of the "responsible person" is typically the person who has primary responsibililty for a premises. This could include facilities managers, owners of premises and managers of buildings.
In multi-tenanted buildings, each tenant has to carry out a fire risk assessment of their premises, and coordinate it with the fire risk assessment that the landlord has carried out of their areas.
In blocks of flats, the fire risk assessment needs to cover the common areas (such as stairs and corridors) but not necessarily the individual flats.
The fire risk assessment has to be carried out to a competent standard. For small buildings, the owner or occupier of the premises is likely to be able to carry the assessement out themselves, as long as they familiarise themselves with the relevant guidance document. However for buildings of any size or complexity, it is best to employ a professional fire risk assessor to carry the assessment out.