With respect to (Al,In,Ga)N epilayers and quantum wells, threading dislocations (TDs) have long been believed to as the principal limiting factor for the internal quantum efficiency of the near-band-edge emission. However, the realization of low TD density GaN and AlN substrates and (Al,In,Ga)N layers enabled investigating the roles of point defects and impurities without interferences by TDs, and vacancy-complexes have been revealed to act as origins of major Shockley- Read-Hall (SRH)-type nonradiative recombination centers (NRCs) in GaN. Accordingly, the concentration of NRCs (NNRC) must be decreased in both optical devices and power-switching electronic devices. Here we show the results of positron annihilation and time-resolved luminescence measurements on n- and p-type GaN, AlN, and Al0.6Ga0.4N alloys to reveal the origins of major intrinsic SRH-NRCs and to obtain their capture coefficients for minority carriers. For unintentionally doped and doped n-type GaN, divacancies comprising of a Ga-vacancy (VGa) and a N-vacancy (VN), namely VGaVN, are assigned as major SRH-NRCs with a hole capture-coefficient (Cp) of 6×10-7 cm3s-1. For Mg-doped ptype GaN epilayers grown by metalorganic vapor phase epitaxy (MOVPE), VGa(VN)2 are assigned as major NRCs with electron capture-coefficient (Cn) of 8×10-6 cm3s-1. For Mg-implanted GaN, VGaVN are the dominant NRCs right after implantation, and they agglomerate into (VGaVN)3 clusters with Cn of 5×10-6 cm3s-1 after high-temperature annealing. Since AlN films grown by MOVPE usually contain vacancy-clusters comprising of an Al-vacancy (VAl) such as VAl(VN)2-3, complexes of a cation-vacancy and a few VNs may be the major NRCs in AlN and Al0.6Ga0.4N alloys.