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In the first half of this dissertation we study certain quotient algebras of preprojective\ud algebras called no-cycle algebras N. These are studied via one-cycle algebras,\ud which are introduced here. Results include detailed combinatorial information on\ud N, and in certain special cases a presentation for N as a quiver with relations.\ud In the second half we consider deformations of coordinate algebras of\ud Kleinian singularities. Results include an explicit presentation for the deformations\ud of a type D singularity.\ud These two themes are tied together at the end by some mainly speculative\ud comments about the role the various objects studied have to play in representation\ud theory

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