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We survey some new and old, positive and negative results on a priori estimate, regularity, rigidity, and constant rankÌýresultsÌýfor special LagrangianÌýand quadratic HessianÌýequations. These equations originate inÌýcalibrated, convex, conformal, and complex geometries among other fields.ÌýUnlike all other order Hessian equations such as the linear (Laplace) and topÌýorder (Monge-Ampere) equations, theÌýregularity/irregularity andÌýa priori Hessian estimatesÌýfor these two equations have not been settled yet in five and higher dimensions.