On Septemper 30th, 2034, the UNIX time function will overflow. More fun stuff that will happen:
On January 1st, 2028, any computers that use the "28 year setback" trick to avoid Y2K will fail to display time properly.
On January 19th, 2038, at 03:14:07 GMT, the seconds counter used for date/time information in UNIX and C and C++ will reach 2,147,483,647, which is the largest number which can be stored as a 32-bit signed integer. This will cause some problems in certain programs which expect the date to never be prior to January 1st, 1970.
On January 1st, 2100, we'll see "Y2.1K"...current PC BIOS clocks will run out of dates. I guess this means they'll no longer get laid either.
On March 1st, 2101, any system which is compliant at this point in time is fully date compliant up until December 31st, 9999.
Source.