Thanks to the success of Proposition 14 in the 2010 election, California now has a nonpartisan blanket primary system for most offices at the state level. (It doesn't affect voting for local offices or for presidential candidates.) The more picturesque term for this is "jungle primaries." What this means is that rather than each party having its own primary, all candidates from all parties for a given office run against each other in a single primary election, and the two with the most votes face off in the general election.
There are various arguments in favor of this scheme, but it has one huge disadvantage: It easily leads to cases in which both candidates in the general election are opposed by a majority of voters. In fact, this has actually happened. In a heavily Democratic district, Democrats split their votes among multiple popular candidates, while Republicans had just two choices to vote for from their own party. The result was that the two Republicans were the ones who made into the general election in the majority-Democratic district. (If you're a Republican this might amuse you, but keep in mind that the reverse situation could easily occur as well.)
As a result, many people in California would like to see a return to traditional partisan primaries, and there's a lot to be said for that. But an alternative I haven't heard discussed would be to let primary voters use what's called "approval voting." In approval voting you cast a vote not just for your favorite choice but for every candidate you'd consider acceptable, however many that might be. The top two candidates, the two declared at least acceptable to the greatest number of voters, would then face off in the general election.
The obvious problem is that there would be no way in the primary for voters to distinguish a candidate they strongly prefer from one they deem a distant but still tolerable second or third or sixth choice. But no voting system is perfect. That's literally true at the level of mathematics, in fact, in that it can be proven that a very brief and reasonable set of desirable attributes cannot be satisfied simultaneously. At least an approval-voting-based jungle primary would be trivially easy to implement and would usually avoid a situation in which the majority voters in the general election face a choice between candidates both of whom they intensely dislike.by