Arizona

process. If the vacancy is in the house, there is a by-election. If the vacancy is in the senate, 45 states allow the governor to appoint a replacement who serves until the next full election. There are caveats about when the senate vacancy actually occurs. See the 17th Amendment. There's nothing in the Constitution that says the Senate appointee has to be the same political party, although some states are now requiring that via legislative action.
House, see Art 1, Sec 2, Clause 4
Senate, see 17th Amendment.

(See Kali's excellent post above, for probably more than you want to know.)