Finding yourself in a situation where you feel you have no choice but to legally evict your adult children from your home is usually reached through a prolonged and often emotionally difficult path. However, once you have reached this decision, the best approach is to try and detach all emotion from the situation, as the question at hand has now become a legal one.
As with any legal matter, the law will vary dependent on your location, so a good first step would be to do some research into where the law in your State/Country stands.
For children aged between 16-18, it may be possible to serve eviction, but you will have to look into the question of what age parental responsibility ends? In the UK parental responsibility ends at the age of 18, but can be reduced to 16 in specific cases (such as if your child gets married, or if determined by court order).
For children that are 18+, things get simpler, but are by no means simple...
In most cases, the law will view your child in the same light as someone who is renting a room in your property, and endow them with all the same rights, regardless of whether any money has exchanged hands at all.
Again, it is best to do research on the laws regarding the eviction process in your local area, most of this information can be obtained fairly easily online or by contacting your local magistrates office. In the majority of cases there will be a recognized procedure for eviction that will initially involve filing paperwork at your local courthouse that will lead to your child being served an official eviction notice.
Once an eviction notice has been served, your child will have the right to a hearing to plead their defense. If there is no legal reason preventing their eviction, then the process will continue and once a motion to evict has been issued (which may involve further paperwork on your part) your child will be given a notice period of generally between 30 and 90 days to vacate the property.
As you can see, the process can be drawn out and will often involve considerable emotional turmoil and difficulty, so if an amicable solution can be reached, then it is always best to pursue that avenue. However, you have the right to a peaceful home life and it is completely correct to stand your ground once every other possible mediation has been exhausted.
One last piece of advice is that it's always good to put as much communication between you and your child down in writing as this helps establish a more tenant/landlord relationship that is more easily defined in court.