"Open Primaries" for Local Elections
Nebraska has nonpartisan elections-- popularly called "open primaries" in many places-- for its one-house legislature. The state lets municipalities decide whether to have party primaries or nonpartisan elections for their own city officials. Currently, the largest city that has party primaries is Fremont, but the city council there is considering changing to nonpartisan elections.
Ballot Access News reports that LB 214 has been introduced in the Nebraska legislature. It would mandate nonpartisan elections for all municipal and county offices.
Mississippi, of course, still has party primaries for municipal and county elections. Since most of our counties decide their elections for county officials in one party's primary, many voters this August will have to choose between voting for state officials or county officials.
This situation could be remedied by the Magnolia State eliminating party primaries for county offices and having all county candidates run in the same election. In each county, all the candidates for county offices would be listed on both the Republican and the Democratic primary ballots. Then, regardless of which party's primary ballot the voter picked for state offices, he would also be able to choose among all the candidates for county offices.
Since the Mississippi legislature is very unlikely to make this change, we citizens will have to do it through a ballot initiative. Hopefully, we will get it accomplished in time for the 2015 state and county elections.
 All candidates, including independents, run in the same election. The top two vote-getters, regardless of party, advance to the runoff.
 Some of Mississippi's smaller municipalities do not hold party primaries in electing their own officials. Rather, all candidates run in the general election, which is a one-round, first-past-the-post election. Thus some officials are elected with less than 50 percent of the vote.