The first step is serving the Section 21 notice or the Section 8 notice, as appropriate.
Section 21 notices must normally provide at least 2 months' notice to the tenant. Section 8 notices can provide between 2 weeks and 2 months of notice - depending on the nature of terms that have been breached by the tenant.
Where a landlord is seeking to evict a tenant who has always been on a periodic (or 'rolling') tenancy (ie a tenant who was never on a fixed-term tenancy), more notice may need to be given. This is because the notice needs to expire at the end of a rental period (ie the notice must expire on the last day of the rental period). For example, take a periodic tenancy that rolls from month to month (ie rent being paid monthly on the first of the month). A landlord serves a section 21 notice partway through the month, on the 15 March. The eviction notice would have to provide 2 months' notice (ie 15 May) but would not be able to end until the last day of the rental period (ie the end of the rental month, here 31 May).
Proof of service notice must be retained, either by filling out a certification of service form (N215) or by writing 'served by [name of landlord] on [date]' on the notice and keeping a copy. If tenants have not vacated by the relevant date, an accelerated possession order can be sought.
Landlords can apply to the court for a possession order if the tenants have not left by the end of the notice period.
Ask a lawyer for more information on applying for a possession order or court proceedings.