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A partial cluster-tilted algebra is the endomorphism ring of a rigid object in a cluster category, and is a generalisation of a cluster-tilted algebra. Unlike cluster-tilted algebras, partial cluster-tilted algebras do not necessarily have global dimension 0, 1 or infinity. Indeed, for each non-negative integer n I will show how to construct a partial cluster-tilted algebra (coming from a cluster category of Dynkin type A) of global dimension n. In studying these algebras, I have been interested in their Auslander-Reiten theory and how it is induced from that of the corresponding cluster category. I will report on the progress so far in this direction and give some idea of other approaches that might be fruitful.