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Examples are constructed to show that the property of F-regularity does not deform. Specifically, we exhibit a three dimensional domain which is not F-regular or even F-pure, but has a quotient by a principal ideal which is F-regular. We show that the invariant subring for the action of the symplectic group on a polynomial ring is, in general, not F-pure. This shows that the socle element modulo an ideal can be forced into the expansion of the ideal in a separable extension, as well as in a linearly disjoint purely inseparable extension. Conditions are examined under which graded rings have Veronese subrings which are F-rational or F-regular. The results obtained give us various techniques of constructing F-rational rings which are not F-regular. For certain classes of local non-equidimensional rings, we prove the conjecture that no ideal generated by a system of parameters can be tightly closed. A new closure operation is constructed, which agrees with tight closure for equidimensional rings, and rectifies the absence of the colon-capturing property of tight closure in non-equidimensional rings. We compute the Frobenius closure and tight closure of certain ideals in diagonal hypersurfaces. This enables us to establish the equality of tight closure and plus closure for these ideals.