ix, 114 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / In this work we prove that the finite W -algebras associated to nilpotent elements in the symplectic or orthogonal Lie algebras whose Jordan blocks are all the same size are quotients of twisted Yangians. We use this to classify the finite dimensional irreducible representations of these finite W -algebras. / Committee in charge: Jonathan Brundan, Co-Chairperson, Mathematics; Victor Ostrik, Co-Chairperson, Mathematics; Arkady Berenstein, Member, Mathematics; Hal Sadofsky, Member, Mathematics; Christopher Wilson, Outside Member, Computer & Information Science