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We construct new examples of Krall discrete orthogonal polynomials, i.e.,orthogonal polynomials with respect to a measure which are also eigenfunctionsof a higher order difference operator. The new examples include the orthogonalpolynomials with respect to the measures obtained from the Meixner measure$\rho_{a,c}$ and Hahn measure $\rho_{a,b,N}$ by removing a finite number oftheir mass points when the parameter $c$ of the Meixner measure and $a$ or $b$of the Hahn measure are positive integers. From the new Krall-Meixner familieswe construct new families of exceptional Meixner and Laguerre polynomials.

New examples of Krall-Meixner and Krall-Hahn polynomials, with applications to the construction of exceptional Meixner and Laguerre polynomials