Adjoint based a posteriori error analysis is a technique to produce exact error representations for quantities of interests that are functions of the solution of systems of partial differential equations (PDE). The tools used in the analysis consist of duality arguments and compatible residuals. In this thesis we apply a posteriori error analysis to the magnetohydrodynamics (MHD) equations. MHD provides a continuum level description of conducting fluids in the presence of electromagnetic fields. The MHD system is therefore a multi-physics system, capturing both fluid and electromagnetic effects. Mathematically, The equations of MHD are highly nonlinear and fully coupled, adding to the complexity of a posteriori analysis. Additionally, there is a stabilization necessary to ensure the so-called solenoidal constraint (div B = 0) is satisfied in a weak sense. We present a novel linearized adjoint system, demonstrate its effectiveness on several numerical examples, and prove its well-posedness. (Joint with Jehanzeb Chaudhry and John Shadid).