The Answer:
You can find your answer in the 17th amendment to the U.S.
Constitution.
Not only did the 17th
amendment establish the process for electing senators, but it
also explained how vacant seats are to be filled.
If a vacancy occurs due to death, resignation, or expulsion the
amendment allows the state legislature to empower the governor to
appoint a replacement. That replacement would hold the seat until the
end of the senator's current term or until a special election could be
held.
The only exception is Arizona. The Grand Canyon State
requires a special election for all vacancies and does not allow any
temporary replacements.
According to the U.S.
Senate's official website, there have been 174 men and women
appointed to fill vacant seats in the Senate. Of that total, 55 have
subsequently been elected to the seat, 55 were defeated, and 64 chose
not to run or were unable to run.
Here is a
list of those appointments.
—The Editors
