Abstract: I will start by defining the Hochschild cohomology of an algebra $A$ with coefficients in an $A$-bimodule $M$. Some examples will be given and special attention will be paid to the case $M=A^*$, the linear dual of $A$. I will then show that Hochschild cohomology is a derived functor, and will use this technique to compute the Hochschild cohomology of noncommutative tori. Following this, I will introduce the cyclic cohomology of an algebra $A$, and will derive Connes' long exact sequence, which provides a powerful tool for computing cyclic cohomology.

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