According to the U.S. Constitution, vacancies are handled as follows:
Article 1, Section 2 (with respect to the House of Representatives): "When vacancies happen in the Representation from any State, the Executive Authority thereof shall issue Writs of Election to fill such Vacancies."
Article 1, Section 3 (with respect to the Senate): ... and if Vacancies happen by Resignation, or otherwise, during the Recess of the Legislature of any State, the Executive thereof may make temporary Appointments until the next Meeting of the Legislature, which shall then fill such Vacancies.
However, this was modified by the 17th Amendment, which provides, "When vacancies happen in the representation of any State in the Senate, the executive authority of such State shall issue writs of election to fill such vacancies: Provided, That the legislature of any State may empower the executive thereof to make temporary appointments until the people fill the vacancies by election as the legislature may direct."
Therefor, it appears, if I read these runes aright, that Representatives must be replaced by special election, but Senators are replaced either by appointment or by special election, depending on the laws of the particular state involved.
Interesting what you find out when you actually research stuff people have told you.