Current unix time
Seconds since Jan 1st 1970 (UTC±0)
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Unix timestamp
Unix time (aka Epoch time or POSIX time) is system to describe time. The value of Unix time is the number of seconds elapsed since the Unix Epoch, which is 00:00:00 UTC on January 1st, 1970.
The Unix time is widely used in many operating systems and Unix timestamp is a very common format to store times. The timestamp is very unambiguous way to represent point of time as it always the same regardless the users time zone.
Year 2038 problem
On many platforms Unix time is represented as a signed integer of 32 bits. That means the minimum representable date is December 13th, 1901 and maximum date is January 19th, 2038. One second after 03:14:07 UTC on January 19th, 2038 the representation of 32 bit signed integer will overflow and will cause problems for many systems.