Problems of estimation of the functional AT ξ = ∞∫−∞T∫0a(s, t)ξ(s, t)dsdt on the unknown values of a random field ξ (s, t) , s ∈ R, t ∈ [0, T ] from observations of the field ξ (u, v)+η (u, v) for (u, v) ∈ R×(R\[0, T ]) are investigated. Formulas are proposed for calculation the mean square errors and spectral characteristics of the optimal linear estimate. The least favourable spectral densities and the minimax-robust spectral characteristics of the optimal linear estimates of the linear functional AT ξ are found for various classes of random fields.