The Constitution specifies that impeachment is an option for “all civil Officers of the United States.” That includes judges, including those on the Supreme Court.
The only Supreme Court justice to be impeached is Samuel Chase as mentioned above, and that impeachment failed in the Senate. But lots of lower federal judges have been successfully impeached and convicted; in the 20th century the tradition has been usually to do this after an ordinary criminal conviction or at least some professional sanction for misconduct. The most recent such case is Samuel Kent who was convicted in a sexual assault case and then impeached when he failed to resign in a timely manner. (He ended up resigning before a Senate trial could commence.)
The last federal judge to be removed by conviction in an impeachment trial is Thomas Porteous in 2010. He was impeached for various mundane improprieties including good old-fashioned bribery, perjury, misstating assets during his bankruptcy, and so on. He was not criminally charged but was convicted in the impeachment trial on four counts and disqualified from holding future office.
All of this applies equally to Supreme Court justices; the fact that no SC justices have been impeached in modern times is probably a reflection of the rather extensive vetting that occurs combined with them being rather small in number. Unlike the presidency, there is no constitutional or theoretical bar to criminally indicting and prosecuting any sitting federal judge, including those on the Supreme Court. Aside from the inevitable media circus, the process would proceed in the usual manner, and Congress could impeach and remove the accused Justice at any point, though in practice they would probably wait until after a criminal conviction.
An impeachment is not reviewable by the courts, though it would certainly be interesting if a substantive question of federal law were raised in a criminal trial involving a Supreme Court justice which ended up being heard by the Court while he was still on it. One would expect that he would recuse himself, though there is no mechanism aside from political pressure to enforce that.