A search for scalar particles and neutralinos such as those predicted by supersymmetric models has been performed using a data sample of 4.4 million hadronic Z(0) decays recorded by the OPAL detector at the e(+)e(-) collider LEP. The production of such particles typically leads to event topologies consisting of an acoplanar pair of jets, or of a mono-jet, accompanied by sizeable missing energy owing to neutrinos and other undetectable neutral particles. Limits are obtained, at the 95% confidence level, on the masses and production rates of scalar particles produced in association with the Z(0). Limits are also placed on neutralino production, for which an additional possible signature is also studied, events containing a single observed photon. Within the Minimal Supersymmetric Standard Model, the mass of the lightest neutralino is found to be larger than 12.5 GeV/c(2) at 95% C.L., provided that tan beta is larger than 1.5.