How does a Therapist end services with Clients?

Sometimes clients come in with one issue to get help with sorting out. Once that is done, the therapist kind of agrees with the client that they have made a lot of progress and they both agree to end therapy... is that true?

Well in the instance that a client has deep seated issues like lifelong depression and anxiety where there are several deep causes and reasons for having these diagnoses... do the therapist and client keep seeing each other until they can no longer meet because of insurance or other outside issues? Say maybe you see the same therapist for years without outside interference... how does that end?