Who pays council tax, the landlord or the tenant?

The tenant, in almost every case, for as long as they are living there under a normal tenancy.

The main exceptions are houses in multiple occupation, where the liability usually sits with the owner, and any period the property is empty between tenancies, when it comes back to you.

Some councils give a short exemption or discount on an empty property, and some charge a premium once it has been empty a long time. It is worth a phone call to the council rather than an assumption, because they set their own rules.