This paper presents a model for computation over arbitrary ordered rings, including the real numbers. The model includes universal machines, partial recursive functions, and NP-complete problems. While reflecting classical theory over the integers, it also highlights the unique mathematical properties of the underlying ring. Key results include: 1. Most Julia sets are not recursively enumerable (R.E.) over the reals, making their complements natural examples of R.E. undecidable sets. 2. An analogue of Cook's NP-completeness theorem is proven over the reals, with the 4-Feasibility problem being NP-complete. 3. Computable functions over the ring are characterized by partial recursive functions. 4. A universal machine over the ring is defined, which is independent of the specific ring. 5. A "Diophantine-like" description of R.E. sets is provided for a class of machines. The paper also discusses the influence of complexity theory on the development of this model, including contributions from Turing, Gödel, Church, and others. It explores the differences between the theories of NP over the reals and over the integers, highlighting the subtle distinctions in decidability and computational complexity.This paper presents a model for computation over arbitrary ordered rings, including the real numbers. The model includes universal machines, partial recursive functions, and NP-complete problems. While reflecting classical theory over the integers, it also highlights the unique mathematical properties of the underlying ring. Key results include: 1. Most Julia sets are not recursively enumerable (R.E.) over the reals, making their complements natural examples of R.E. undecidable sets. 2. An analogue of Cook's NP-completeness theorem is proven over the reals, with the 4-Feasibility problem being NP-complete. 3. Computable functions over the ring are characterized by partial recursive functions. 4. A universal machine over the ring is defined, which is independent of the specific ring. 5. A "Diophantine-like" description of R.E. sets is provided for a class of machines. The paper also discusses the influence of complexity theory on the development of this model, including contributions from Turing, Gödel, Church, and others. It explores the differences between the theories of NP over the reals and over the integers, highlighting the subtle distinctions in decidability and computational complexity.