Until recently, very little was known about the higher algebraic K-theory of anything but finite fields. This paper takes advantage of recent work connecting K-theory to cyclic homology, and computes the higher algebraic K-theory of seminormal curves in characteristic zero, assuming the K- theory of smooth curves and fields as given. These are the first calculations of this kind, and establish a new level of computational knowledge about K-theory. For curves with cusp-like singularities, a calculation is given, assuming the validity of the conjecture that double relative K-theory is isomorphic to double relative cyclic homology. The first half of the paper is of independent interest. In it, a new method for calculating the Hochschild and cyclic homology of graded algebras is developed. Together with the appendix on localization and analytic isomorphisms in Hochschild and cyclic homology, this enables the authors to calculate the cyclic homology of many new rings.