OD&D: No extra weight, critical hits and fumbles were house rules, third party supplements like Arduin might have them.
AD&D: Same. Although the Saving Throw rules mention that a nat 1 always fails regardless of modifiers, which can very much matter at high levels with magic items.
B/X: First D&D edition to make a nat 1 auto-miss and a nat 20 auto-hit on an attack roll. No impact noted on saves.
2E: Nat to hit 20 auto-hits, nat 1 auto-misses. Critical hits and fumbles are optional rules.
3E: First D&D edition to make critical hits official. 1s on attacks auto-miss, 20s threaten a critical, which you have to roll to confirm. 1s and 20s are nothing special on saves, but since Save DCs go up and up with higher level foes, the 1E auto-fail on a 1 rule really isn't needed. Critical success and failures on skill checks is an optional rule. I had forgotten this last being there! 3.5 DMG p92.
4E: 20s on attacks auto-hit, and if they would normally hit the AC they're critical. Critical hits are official, fumbles not present, 1s and 20s mean nothing special on a skill check. Saving throws redefined and only give specials on a Death Save of 20, if you have any healing surges left you get to spend one (healing from 0 instead of negatives) and regain consciousness.
5E: 20s to hit are crits, 1s are auto-misses, on death saves 20s restore one HP, 1s fail two death saves, on other saves and on skill checks 1s and 20s are nothing special.