Simultaneous　optimal　estimation　in　linear　mixed　models　is　considered.　A　necessary　and　sufficient　condition　is　presented　for　the　least　squares　estimator　of　the　fixed　effects　and　the　analysis　of　variance　estimator　of　the　variance　components　to　be　of　uniformly　minimum　variance　simultaneously　in　a　general　variance　components　model.　That　is,　the　matrix　obtained　by　orthogonally　projecting　the　covariance　matrix　onto　the　orthogonal　complement　space　of　the　column　space　of　the　design　matrix　is　symmetric,　each　eigenvalue　of　the　matrix　is　a　linear　combinations　of　the　variance　components　and　the　number　of　all　distinct　eigenvalues　of　the　matrix　is　equal　to　the　the　number　of　the　variance　components.　Under　this　condition,　uniformly　optimal　unbiased　tests　and　uniformly　most　accurate　unbiased　confidence　intervals　are　constructed　for　the　parameters　of　interest.　A　necessary　and　sufficient　condition　is　also　given　for　the　equivalence　of　several　common　estimators　of　variance　components.　Two　examples　of　their　application　are　given.