Error estimation and adaptivity by stress reconstruction in elasticity are studied in this work. The problem of elasticity is solved by a displacement-pressure approach, where the linear elastic case and non-linear elastic case are considered and discussed separately. First a general framework for weakly symmetric stress reconstruction is estaablished. The general framework is extended to linear elasticity and non-linear elasticity, e.g. hyperelasticity. The weak symmetry constraint differs in the linear and non-linear case and have to be treated accordingly. By localizing this global reconstruction scheme compatiability conditons for the local problen needs to be satisfied. The appearance and treatment of mentioned compatibility conditons are discussed, such that the presented apprach leads to a truly equilibrated stress reconstruction in the linear and non-linear case. Afterwards in the linear elastic case an a-posteriori error estimatior is presented for which proofs for efficiency and reliabilitiy are provided. The errror estiamtor is examined in several numerical  examples. In addition key features of the reconstructed stress to the approximated stress by the displacement-pressure are tested. and discussed. In the non-linear elastic case the reconstructed stress is used as an error indicator and examined accordingly.