Within mean-field theory we study wetting of elastic substrates. Our analysis is based on a grand canonical free energy functional of the fluid number density and of the substrate displacement field. The substrate is described in terms of the linear theory of elasticity, parametrized by two Lamé coefficients. The fluid contribution is of the van der Waals type. Two potentials characterize the interparticle interactions in the system. The long-ranged attraction between the fluid particles is described by a potential w(r), and v(r) characterizes the substrate-fluid interaction. By integrating out the elastic degrees of freedom we obtain an effective theory for the fluid number density alone. Its structure is similar to the one for the wetting of an inert substrate. However, the potential w(r) is replaced by an effective potential which, in addition to w(r), contains a term bilinear in v(r). We discuss the corresponding wetting transitions in terms of an effective interface potential ω(ℓ), where ℓ denotes the thickness of the wetting layer. We show that in the case of algebraically decaying interactions the elasticity of the substrate may suppress critical wetting transitions, and may even turn them first order.