<p>In this thesis we focus on identically self-dual matroids and their minors. We show that every sparse paving matroid is a minor of an identically self-dual sparse paving matroid. The same result is true if the property sparse paving is replaced with the property of representability and more specifically, F-representable where F is a field of characteristic 2, an algebraically closed field, or equal to GF(p) for a prime p = 3 (mod 4). We extend a result of Lindstrom [11] saying that no

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... ally self-dual matroid is regular and simple. We assert that this also applies to all matroids which can be obtained by contracting an identically self-dual matroid. Finally, we present a characterisation of identically self-dual frame matroids and prove that the class of self-dual matroids is not axiomatisable.</p>