This paper classifies and constructs explicitly all the irreducible representations of affine Hecke algebras of rank two root systems. The methods used to obtain this classification are primarily combinatorial and are, for the most part, an application of the methods used in [Ra1]. I have made special effort to describe how the classification here relates to the classifications by Langlands parameters (coming from p-adic group theory) and by indexing triples (coming from a q-analogue of the Springer correspondence). There are several reasons for doing the details of this classification: (a) The proof of the one of the main results of [Ra1] depends on this classification of representations for rank two affine Hecke algebras. Specifically, in the proof of Proposition 4.4 of [Ra1], one needs to know exactly which weights can occur in calibrated representations. The reason that this naturally depends on a rank two classification is outlined in (d) below. (b) The examples here illustrate (and clarify) results of [Ra1], [KL], [CG], [BM], [Ev], [Kr], [HO1-2]. Much of the power of the combinatorial methods which are now available is evident from the calculations in this paper, especially when one compares with the effort needed in other sources (for example [Xi], Chapt. 11). (c) The explicit information here can be very useful for obtaining results on representations o
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