DAT and IPFS are attempts to make the web way more peer-to-peer.
I don't know as much about IPFS, but I'm learning.
DAT forms the bones of the "web without the internet" project I've been working on (http://ajroach42.com/steps-towards-a-web-without-the-internet/)
Unfortunately, DAT currently only works on (relatively modern) desktops. Eventually, it should also work on mobile devices, but until then... it's not an ideal solution.
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