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We present astrometric very long baseline interferometry (VLBI) studies ofAGB stars. To understand the properties and evolution of AGB stars, distancesare an important parameter. The distribution and kinematics of theircircumstellar matter are also revealed with the VLBI method. We used the VERAarray to observe 22\,GHz H$_2$O masers in various subclasses of AGB stars.Parallaxes of the three OH/IR stars NSV17351, OH39.7$+$1.5, IRC$-$30363, andthe Mira-type variable star AW~Tau were newly obtained. We present thecircumstellar distribution and kinematics of H$_2$O masers around NSV17351. Theabsolute magnitudes in mid-infrared bands of OH/IR stars with very longpulsation periods were investigated and a period-magnitude relation in the WISEW3 band, $M_{\mathrm{W3}} = (-7.21\pm1.18)\log P + (9.25\pm3.09)$, was foundfor the Galactic AGB stars. The VLBI is still a powerful tool for parallaxmeasurements of the Galactic AGB stars surrounded by thick dust shells.

Implication of the period-magnitude relation for massive AGB stars and its astronomical applications