Complex cryogenic plasma is theoretically and experimentally studied. In a cryogenic plasma the Debye length becomes smaller and the shielding length becomes comparable to a dust size. Dust charging in a cryogenic plasma is examined in detail by considering the polarization effect on a dust particle due to charged plasma particles. Electrons are found to be attracted to a dust particle when they come closer to a dust particle within a distance from the surface of a dust particle given by Δr = a ((ε − 1)exp(a/λD)/2|Zd|(1 + a/λD)(2ε + 3))1/2, where a is the radius of a spherical dust particle, ε is the dielectric constant of the dust particle, Zd is the charge state of a dust particle and λD is the Debye length. Microscopic electron collection flux balancing with desorption electron flux determines the dust charge. Negatively charged dust particles form two-dimensional crystal structures and the displacement from the equilibrium positions shows normal modes with a dispersion relation. Experimental study determines dust charges on a cryogenic temperature.