he reduced basis (RB) methods are proposed here for the solution of parametrized equations in linear elasticity problems. The fundamental idea underlying RB methods is to decouple the generation and projection stages (offline/online computational procedures) of the approximation process in order to solve parametrized equations in a rapid, inexpensive and reliable way. The method allows important computational savings with respect to the classical Galerkin-finite element method, ill suited to a repetitive environment like the parametrized contexts of optimization, many queries and sensitivity analysis. We consider different parametrization for the systems: either physical quantities --to model the materials and loads-- and geometrical parameters --to model different geometrical configurations--. Then we describe three different applications of the method in problems with isotropic and orthotropic materials working in plane stress and plane strain approximation and subject to harmonic loads.