Most routers (including humans) haven't even figured out that you can run traces in more than just 8 directions :) I quite like autorouted boards in TopoR for this reason, but it's hardly cutting edge; been around for decades.

I hear Altium has some pretty decent autorouting. The problem here is that by the time you've set up all the grouping, constraints and settings to get an auto route that works well, you probably could have just routed the board yourself. So I've never used it once. There's still no magic button you can click once to make a full routed board. As far as I know, autorouting for all the big players is still based on tried and true algorithms like A-star. They might be getting a little better year over year, but not so much so that anyone noticed.

I have seen quite a few improvements in manual routing though. It seems like the manual routing tool has gotten better at pushing traces out of the way, and if you move a routed component you might not need to redraw all the traces going to it anymore.

> Slightly tangential, but what's the state of the art in auto routing and placement these days?

I always recommend quilter.ai: https://www.quilter.ai/ - they "eat" Kicad or Altium files, and produce autorouting and/or autoplacement. I tried them about 9 months ago and made a couple of boards that worked well.