Thanks to their low galactic latitude and fast proper motion, the components of the nearest triple system Alpha Centauri (A, B and Proxima) approach distant background stars at small angular separations relatively frequently. Close approaches between Proxima and distant stars were observed with the HST and the VLT in 2016, enabling the estimation of its gravitational mass. The Alpha Cen A and B components are too bright to be measured by Gaia, but precise astrometry with ALMA and VLT/NACO enable us to accurately predict future lensing event configurations. We will present the conjunctions that will occur within the next decade (particularly the S5 star approach in 2028) and a preliminary report on our GRAVITY observations of the first conjunction that occurred in April 2021. These conjunctions offer promising opportunities to search for exoplanets in the Alpha Cen system, measure the gravitational masses of the stars, as well as dramatically improve the accuracy of the space velocity of the system. The latter is important, e.g., to assess the feasibility of a ballistic mission to Alpha Centauri such as the Breakthrough Starshot concept.