32bit systems will stop working. The Unix timestamp, which increases by 1 every second and started the first second of 1970, will reach the max of 32 bit integers. Bad things will follow.
This has already been patched on all 64 bit OSes though - whatever 32 bit systems are still in existence in another 15 years will just roll their dates back 50 years and add another layer of duct tape to their jerry-rigged existence
And now that every time library has been updated, we're safe until our grandchildren reimplement those bugs in a language that has not yet been invented.
I went to uni in the mid 90s when Y2K prep was all the rage, went back to do another degree 20 years later. It was interesting to see the graffiti in the CS toilets. Two digits up to about 1996, four digits for a decade, then back to two.