The goal of the paper is to define Hochschild and cyclic homology for bornological coarse spaces, i.e., lax symmetric monoidal functors XHHG and XHCG from the category GBornCoarse of equivariant bornological coarse spaces to the cocomplete stable infinity-category Ch(infinity) of chain complexes reminiscent of the classical Hochschild and cyclic homology. We investigate relations to coarse algebraic K-theory X K-G and to coarse ordinary homology XHG by constructing a trace-like natural transformation X K-G -> XHG that factors through coarse Hochschild (and cyclic) homology. We further compare the forget-control map for XHHG with the associated generalized assembly map.