Their current favorite is to hijack a third-party mail server. The spammers, in effect, launder their junk email through third-party relays to slip through the spam filters. Spammers use relays to increase the number of messages they can spew. A lowly PC sitting at the end of a phone line can only pump out a limited number of messages. Further, if the spammer can relay through several mail servers in parallel, they can flood the net with extraordinary amounts of junk mail.
Spammers can hide behind third-party relays.
If a spammer sends junk email directly, network managers can trace back the connection and deal with the problem. If, instead, the spammer relays the mail, they may be able to obscure their identity. Even if the spammer can't hide completely, they will deflect a significant portion of the complaints away from themselves and towards the administrators of the hijacked host. In fact, many spammers forge bogus mail headers to encourage this misdirection. Spammers hijack mail servers because it greatly increases the amount of spam they can deliver, all at no cost to them.