The present sexagenary cycle began in Gregorian year 1984.
Wikipedia identifies 1984 as a jiazi year, the first term of the current 60-year cycle. The cycle counts years through combinations of ten Heavenly Stems and twelve Earthly Branches; because their least common multiple is 60, the sequence returns to its starting combination every six decades.
The cycle’s New Year does not necessarily begin on January 1. In the traditional Chinese calendar, the count changes at Lunar New Year. In some astrological systems, the year is instead considered to begin at Lìchūn, the first solar term, usually near February 4. Thus, “1984 began the cycle” refers to the traditional calendrical year rather than an uncomplicated January 1 boundary.
The next cycle beginning after the present one is scheduled for 2044. The year immediately before 1984, 1983, was the final term of the preceding cycle. The sexagenary system was used for years later than it was for days: year-counting began around the middle of the 3rd century BC and became widespread during the Western Han dynasty.