Sometimes, when a defendant gets a complaint, the defendant wants to say: What happened to the plaintiff isn’t my fault. It is the fault of another party. The defendant might raise this argument as an affirmative defense, or the defendant might want to bring that other party into the plaintiff’s lawsuit. The defendant can do this by bringing a third-party claim in the answer. In doing so, the title of the answer changes from Answer to Answer and Third-Party Complaint. As is done with a counterclaim, the defendant includes a third-party claim in the answer by essentially including complaint-like allegations in the body of the answer.
The defendant serves the Answer and Third-Party Complaint on the plaintiff in the same way the defendant would serve a simple answer, that is, usually by mail. The defendant serves the Answer and Third-Party Complaint on the party who would be called the third-party defendant in the same way the plaintiff served the complaint on the defendant, that is, usually using a process server, and complying with all of the elaborate rules governing the service of a complaint on a defendant.
Although this is rare, a third-party defendant can bring a claim against a fourth-party defendant, and a fourth-party defendant against a fifth-party defendant. Theoretically, this can go on forever, but I doubt it this happens often.
When the third-party defendant serves its third-party answer on the defendant, the third-party defendant may also bring cross-claims against the plaintiff, in the same way a defendant can bring a counterclaim against the plaintiff. If there are multiple defendants in a lawsuit, each defendant can bring cross-claims against some or all of the other defendants. Fourth-party and fifth-party defendants can also bring cross-claims against any other party in the lawsuit. The plaintiff may also bring cross-claims. Although it sounds as if it can get pretty messy and confusing, in reality, this type of spider-web of claims is very rare. I have been involved in thousands of cases, and I have seen something like this only a handful of times, maybe only once or twice..