Single crystals ofRMg2Cu9 (R = Y, Ce-Nd, Gd-Dy, Yb) were grown using a high-temperature solution growth technique and were characterized by measurements of room-temperature x-ray diffraction, temperature-dependent specific heat, and temperature- and field-dependent resistivity and anisotropic magnetization. YMg2Cu9 is a nonlocal- moment-bearing metal with an electronic specific heat coefficient, γ ∼ 15 mJ/mol K2. Yb is divalent and basically non-moment-bearing in YbMg2Cu9. Ce is trivalent in CeMg2Cu9 with two magnetic transitions being observed at 2.1 K and 1.5 K. PrMg2Cu9 does not exhibit any magnetic phase transition down to 0.5 K. The other members being studied (R = Nd, Gd-Dy) all exhibit antiferromagnetic transitions at low temperatures ranging from 3.2 K for NdMg2Cu9 to 11.9 K for TbMg2Cu9.Whereas GdMg2Cu9 is isotropic in its paramagnetic state due to zero angular momentum (L = 0), all the other local-moment-bearing members manifest an anisotropic, planar magnetization in their paramagnetic states. To further study this planar anisotropy, detailed angular-dependent magnetization was carried out on magnetically diluted (Y0.99Tb0.01)Mg2Cu9 and (Y0.99Dy0.01)Mg2Cu9. Despite the strong, planar magnetization anisotropy, the in-plane magnetic anisotropy is weak and field-dependent. A set of crystal electric field parameters are proposed to explain the observed magnetic anisotropy.