When will the next series of mutual eclipses of Jupiter's moons begin? When I was young (I won't tell you when) I saw one of the four Galilean moons of Jupiter suddenly disappear while watching through a small refractor. It wasn't a coincidence, I'd seen the prediction in some announcements in Sky and Telescope and it happened like clockwork. The orbital plane of the Galilean moons had rotated so that it included the Sun, and the moons were Eclipsing each other, or so I remember.

Will this happen again? If so, when will they start, and where can I find predictions of individual eclipses?
Mutual eclipses and occulations of Jupiter's moons occur during Jovian Equinox, twice in each orbit of Jupiter, and so roughly every 6 years.

The last series of events was in 2015, so the next will start in 2021. There will be a series of events between January and August. 

The site http://lnfm1.sai.msu.ru/neb/nss/nsszph517he.htm can generate ephemerides for these events.