The talk overviews recent research by the authors on the development of second-order consistent and stable lattice Boltzmann formulations to solve elastostatics and elastodynamics problems. The first proposed scheme [1] solves the quasi-static equations of linear elasticity using a collision operator with multiple relaxation times. In contrast to previous works, our formulation solves for a single distribution function with a standard velocity set and avoids any recourse to finite difference approximations. As a result, all computational benefits of the lattice Boltzmann method can be used to full capacity. The second proposed scheme [2] solves the equations of linear elastodynamics. The construction of the numerical scheme uses an equivalent first-order hyperbolic system of equations as an intermediate step, for which a vectorial lattice Boltzmann formulation is introduced. Both schemes are systematically derived using the asymptotic expansion technique and stability is assessed for elastostatics with von Neumann analysis, whereas in elastodynamics we exploit the notion of pre-stability structures to prove stability for an arbitrary combination of material parameters under a CFL-like condition. Boundary formulations for various cases are proposed. The first steps towards the solution of nonlinear problems are discussed.