We live in a republic. A so called representative democracy. The way we achieve this is by dividing the nation up into districts, and having each district elect one representative. The idea being that the elected official, in some way, represents all the residents of that district. Though that's the ideal, it's far from the truth. The truth of the matter, if you consider it for just a moment, is that a representative only represents those who vote for him. Every time someone says something like, "It isn't my fault, I didn't vote for <blank>," you know that person doesn't feel represented by that politician. Under this conception of representation, the system we use is absolutely terrible.
In theory, as few as 25% of voters in a shifting district system with two parties can run the nation. Simplified example - every voter belongs to one of two parties, and votes that way reliably. Set up the districts so that half of the districts are 100% one way, and the rest are 50-50. In the lawmaking bodies both populations will achieve equal representation. In other words, a party that represents the views of 25% of the population is competitive with a party that represents 75%, when they shouldn't even be close. The situation is ameliorated some by the presence of swing voters, and the imperfect loyalty of even partisan voters. It's made far worse, however, by low voter turnout.
Static districting is not an option, so if we eliminate districting, what options do we have? The most obvious option is to have the entire state vote for the entire slate of reps. If everyone has as many votes as there are reps to elect, then the delegation will tend to be a pretty uniform. In the worst case, about 50% of the population runs the show. It will also tend not to do much better than that, either. It strikes me as likely that the result will be similar for all voting systems that permit multiple votes for multiple offices. That leaves each person one vote for for one congressional candidate in their state, and the top X candidates go to congress. How would this system work? Well, it can work well, but is so full of annoying problem cases that can be disastrous. For instance, several similar parties split the majority of the vote, and a minority party takes the election. Another similar situation happens when you have an extremely popular candidate get a lion's share of the vote.
There is actually a way to get around all of the above problems. There is, of course, a trade-off involved. In this case, the tradeoff would be a reduction in politically problematic situations in exchange for increased logistical requirements. I like to call the system "Democracy by Proxy." The key difference between Democracy by Proxy and a Representative Democracy is that in the latter you vote for someone to represent you in a legislative body of equals, in the former you vote for, or designate, someone to cast your votes in a legislative body that technically includes all of the people. The key difference is, of course, that a Proxy gets to cast as many votes as she receives in an election. A Representative gets one vote no matter how many votes he received. This actually eliminates several logistical problems of the system we use now right off the bat. For starters, the size of a state's delegation no longer determines its political influence. Thus, there's no longer any reason to pre-determine the size of a state's delegation. The census would also no longer be nearly as politically sensitive, though still necessary for reasons I'll go into later.
I said that the tradeoff moves most of the problems from political to logistical. The most obvious logistical problem is keeping a delegation's size manageable. Thus far, in the system described millions of people could vote for themselves, and take their one vote into the legislative body. While it would be nice if this were practicable, it simply won't work. So, as a practical matter, delegation sizes have to be limited somehow. Whether that limit should be a straight up limit on delegate number, or a minimum number of votes necessary to be sent as a delegate is debatable. Using a "top X" Proxies system will lead to the same split vote anomalies that made just eliminating districts unattractive (so far, we've only addressed the "lions share of the vote" problem). In the spirit of maximizing the amount of the population represented, requiring a minimum number of votes instead of a maximum number of proxies is preferable. Technically, the two systems can be made to perform the same at any given time, but over time one will grow with the population and the other will not. What's more, it's amenable to another tweak to help maximize representation that would not, on balance, help the delegation limited system.
In a minimum vote Democracy by Proxy it is possible to increase the overall representation by allowing candidates who did not receive enough votes to gather in to groups of their own choosing, and turn their votes over to another candidate. The idea being that they are electing a proxy for their group. This should take care of any split vote problems, especially in situations where the minimum vote is set high. I have some reservations about possible problems with this tweak. Again, arising from concerns about consolidation of power in too few hands. Especially if Proxies that are above the cutoff are allowed to participate. This should be alleviated if the Proxies are allowed to withdraw support at any time. There will have to be a lower cutoff to participate in this system, too, because otherwise there's no excuse to exclude the voters from withdrawing support from Proxies and reassigning it to another at any time. They could just vote for themselves, losing anonymity but gaining more control over their vote. That not actually a bad idea in some respects, but it's a complete logistical nightmare.
To keep the logistical challenge of running the House to a minimum one can take advantage of the fact that the proxies are not all equal. For instance, requiring that a Proxy have a minimum number of votes backing him up to get floor time (ie if he doesn't have enough on his own, he needs to seek backing from other proxies), and a minimum number of votes to introduce legislation for consideration. These solutions, interestingly enough, don't necessarily have to be limited to a Democracy by Proxy system. They can also help in the case where we simply opt to expand the House of Representatives as it is.
Ballot access is another issue that can lead to logistical headaches for the voter. Requiring every candidate to seek state wide ballot access seems excessive, and would lead to a very long and confusing list of candidates on the ballot. Given that, it would be better if candidates have the option of gaining access district by district (the districts could be counties, or maybe not) or in the full state all at once (depending on the campaign they want to run, naturally). If write-ins are required to be accepted, the districts server more as a guideline and indication of where the candidates intend to concentrate their efforts, and thus redistricting shouldn't be a huge problem.
Democracy by Proxy also affords candidates some interesting political opportunities that aren't available as a Representative. First, a Proxy can define her constituency however makes sense to her (geographically, ideologically, etc). Second, a Proxy has the advantage of being able to split his own votes on an issue. Say, for instance, that a Proxy doesn't stand behind a bill wholeheartedly. He can evenly split his votes, with a small number of votes in excess of the other going the way he's leaning (or just cast some of his votes - same difference). This gives the ability to express nuance far in excess of a straight up or down fashion.
Democracy by proxy also has at least one unique problem that must be addressed. That problem is - can a Proxy cross state lines? My initial response is that it sounds just fine to me. The problem that arises in this instance is that it can lead to a one, or some other very small number, proxy legislature. It's a long shot, but one that I would prefer to expressly forbid simply because that's too much power to have in just a few hands. Thus, I would insist on limiting proxies to receiving votes from one, two, or at the very most three states. It's either that, or limit the number of votes any one Proxy can represent. Frankly, I prefer the former because it's more clear before you vote whether or not a Proxy can receive your vote, especially given that ballot access controls can reflect this reality. Granted, this problem is a long shot. A coordinated team (ie in the same party) of localized candidates who can focus on small areas should be able to outperform a diffused candidate almost every time. Even so, under extraordinary circumstances it could happen.
The largest problem with such a system is political in nature, but logistical in solution. In a Democracy by proxy, the temptation to increase one's vote count is tremendous and ever-present. Normally, if an election isn't close, voter fraud won't be able to change the outcome. In this system, increasing the vote count is always desirable. This presents an advantage of the system in that politicians will try to maximize participation using legitimate means. The temptation to use illegitimate means will also be almost impossible to resist. Because of that, if this system is to be married to anonymous balloting, the vote counting system must be transparent, accurate, and open. In a word, bulletproof. How this is to be achieved is a technical question that has multiple solutions. The census should be continued, for instance, because knowing the total population of a district gives a maximum on how many votes can come out of that district.