  In arXiv:1209.0038 we constructed topological triangulated categories C_c as
stable categories of certain topological Frobenius categories F_c. In this
paper we show that these categories have a cluster structure for certain values
of c including c=pi. The continuous cluster categories are those C_c which have
cluster structure. We study the basic structure of these cluster categories and
we show that C_c is isomorphic to an orbit category D_r/F_s of the continuous
derived category D_r if c=r pi/s. In C_pi, a cluster is equivalent to a
discrete lamination of the hyperbolic plane. We give the representation
theoretic interpretation of these clusters and laminations.
