The earlier generation of paperless electronic voting systems, adopted in the early 2000's, have been widely studied and have been found to have significant security flaws (examples: California "top to bottom" review in 2007, Ohio EVEREST 2007). (I was one of the co-authors on the California review.)

As a consequence, all the new voting machines involve paper in one form or another. The two most popular forms are ballot marking devices, which have some sort of computer interface and produce a printed ballot, and hand-marked paper ballots, which are typically scanned by a computer, often bolted to the top of the ballot box ("precinct count optical scanner").

The magic of a risk limiting audit (the topic of this thread!) is that it provides an efficient process where a post-election audit can prove, to a desired level of statistical confidence, that any errors in the electronic tabulation are small enough that they don't change the announced winner of a contest.

So, RLAs let us have the efficiency benefits of computers, while still having the security properties that we want from hand tallies, without requiring the slow (and error-prone) process of hand counting.