While the United States Constitution does set parameters for the election of federal officials, state law, not federal, regulates most aspects of elections in the U.S., including primaries, the eligibility of voters (beyond the basic constitutional definition), the running of each state 's electoral college, as well as the running of state and local elections. All elections -- federal, state, and local -- are administered by the individual states.
who are responsible for running primary general and special elections
individual states