In 1990, Downey won the Hamilton Research Award from the
Royal Society of New Zealand. In 1992, Downey won the Research Award of the New Zealand Mathematical Society "for penetrating and prolific investigations that have made him a leading expert in many aspects of recursion theory, effective algebra and complexity". In 1994, he won the New Zealand Association of Scientists Research Award, and became a fellow of the Royal Society of New Zealand in 1996. In 2006, he became the first New Zealand-based mathematician to give an Invited Lecture at the
International Congress of Mathematicians. He has also given invited lectures at the International Congress of Logic, Methodology and Philosophy of Science and the ACM Conference on Computational Complexity. He was elected as an
ACM Fellow in 2007 "for contributions to computability and complexity theory", becoming the second ACM Fellow in New Zealand, and in the same year was elected as a fellow of the
New Zealand Mathematical Society. In 2010 he won the Shoenfield Prize (for articles) of the
Association for Symbolic Logic for his work with Denis Hirschfeldt, Andre Nies, and Sebastiaan Terwijn on
randomness. In 2011, the Royal Society of New Zealand gave him their
Hector Medal "for his outstanding, internationally acclaimed work in recursion theory, computational complexity, and other aspects of mathematical logic and combinatorics." In 2012, he became a fellow of the
American Mathematical Society. In 2013, he became a Fellow of the
Australian Mathematical Society. In 2014, he was awarded the
Nerode Prize from the
European Association for Theoretical Computer Science, jointly with
Hans Bodlaender, Michael Fellows, Danny Hermelin,
Lance Fortnow and Rahul Santhanam for their work on
kernelization lower bounds. In October 2016, Downey received a distinguished
Humboldt Research Award for his academic contributions. With Denis Hirschfeldt, Downey won another Shoenfield Prize from the Association for Symbolic Logic, this time the 2016 book prize for
Algorithmic Randomness and Complexity. In 2018, Downey delivered the
Gödel Lecture of the Association for Symbolic Logic, titled
Algorithmic randomness, at the European Summer Meeting at Udine, Italy. The same year, Downey was awarded the
Rutherford Medal, the highest honour awarded by the Royal Society of New Zealand, "for his pre-eminent revolutionary research into computability, including development of the theory of parameterised complexity and the algorithmic study of randomness." In 2022, Downey was awarded the New Zealand Association of von Humboldt Fellows Research Award for research over the preceding five years. In 2023, Downey was awarded the S. Barry Cooper Prize from the Association for Computability in Europe. This award is awarded every two to three years "to a researcher who has contributed to a broad understanding and foundational study of computability by outstanding results, by seminal and lasting theory building, by exceptional service to the research communities involved, or by a combination of these." In 2024, Downey was awarded the New Zealand Mathematics Society Kalman Prize "for a single publication of original research, which may be an article, monograph or book, having appeared within the last 5 calendar years: 2019-2024". This publication was the monograph "A Hierarchy of Turing Degrees" published in the Annals of Mathematics Studies, jointly written with Noam Greenberg. ==Scottish Country Dancing==