Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin consists of invited expository articles and new developments extending from Yuri I. Manin outstanding contributions to algebra, algebraic geometry, number theory, algorithmic complexity, noncommutative geometry and mathematical physics. Among Manin numerous achievements are the proof of the functional analogue of the Mordell conjecture, the theory of Gauss--Manin connection, the proof with V. Iskovskikh of the nonrationality of smooth quartic threefolds, the theory of $p$-adic automorphic functions, and the theory of quantum computations. Contributors in the first volume include: K. Behrend V.G. Berkovich J.-B. Bost P. Bressler D. Calaque J.F. Carlson A. Chambert-Loir E. Colombo A. Connes C. Consani A. Dabrowski C. Deninger I.V. Dolgachev S. K. Donaldson T. Ekedahl A.-S. Elsenhans B. Enriquez P. Etingof B. Fantechi V.V. Fock E.M. Friedlander B. van Geemen G. van der Geer E. Getzler A.B. Goncharov V.A. Iskovskikh J. Jahnel M. Kapranov E. Looijenga M. Marcolli B. Tsygan E. Vasserot M. Wodzicki