In Australia, for most purposes we still use paper ballots. (There are a few exceptions - ACT territory elections have *optional* computer-based voting, and NSW state elections have an *optional* online voting system for some absentee or disabled voters.)
On election night, officials at every polling place - who are required to sign a declaration, under penalty, that they are not politically active - do an initial hand count of first-preference votes (yes, we have IRV and STV ballots here) and the votes for the top two front runners. These are the numbers that make their way to the internet in a matter of minutes and are used for the election night media coverage - but they actually have no legal significance at all, they're basically purely for the media coverage.
The real counting happens the week after election day, when all ballots are transported to the local electoral office for counting. For elections that use IRV ballots (e.g. the federal House of Representatives), the ballots are all hand counted. For STV ballots (e.g. the federal Senate), they do use computer based counting, however the paper ballots are retained and a hand count can be done if necessary. If there are any issues that arise, the Returning Officer has the discretion to order a recount as necessary, without necessarily needing court orders or anything like that.
The *entire process* - opening the polls, conducting the polling, closing the polls, the first count, the second count, and any recounts - takes place in front of candidate-appointed scrutineers (not quite as good as being public, but it's close enough). Every candidate can appoint scrutineers to witness the whole process and make objections.
And this is how Australia has elections that are virtually unchallengeable - for a typical federal election, there will usually be at most one serious dispute, and only in districts with the tiniest of margins where they need a judge to make the final decision. Heck, we're experimenting with computer-based and internet-based voting systems, and no-one's raising concerns because the Electoral Commission has such a high reputation for integrity and accuracy.