| Summary: | <p>We investigate several types of Nakajima quiver varieties and their connection to the Kleinian singularities 𝐂<sup>2</sup>/Γ.</p>
<p>Such quiver varieties have many good properties: they are for instance irreducible and have symplectic singularities, in particular they are normal.</p>
<p>The first part of this thesis introduces Nakajima quiver varieties, together with the necessary background on projective Deligne-Mumford stacks and framed sheaves. Following that, we prove that the punctual Hilbert schemes (when taken with their reduced scheme structures) Hilb<sup>n</sup>(𝐂<sup>2</sup>/Γ) are examples of Nakajima quiver varieties.</p>
<p>We then show that there is a type of 'orbifold Quot scheme' generalising the Hilbert scheme, which can also be identified with a Nakajima quiver variety.</p>
<p>Following this, we investigate another generalisation of the Hilbert scheme: that of a moduli space of framed sheaves on a certain stack compactifying 𝐂<sup>2</sup>/Γ. We show that this moduli space exists as a quasiprojective scheme. We are unable to show that it is isomorphic to a Nakajima quiver variety, but we show that it carries a canonical morphism to a Nakajima quiver variety, and this morphism is a bijection of closed points.</p>
<p>We end by sketching some potential further directions of investigation.</p>
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