The Third Edition Of This Timely, Definitive, And Popular Book Continues To Pursue The Question: What Is The Most Efficient Way To Pack A Large Number Of Equal Spheres In N-dimensional Euclidean Space? The Authors Also Continue To Examine Related Problems Such As The Kissing Number Problem, The Covering Problem, The Quantizing Problem, And The Classification Of Lattices And Quadratic Forms. Like The Previous Edition, The Third Edition Describes The Applications Of These Questions To Other Areas Of Mathematics And Science Such As Number Theory, Coding Theory, Group Theory, Analog-to-digital Conversion And Data Compression, N-dimensional Crystallography, Dual Theory And Superstring Theory In Physics. Of Special Interest To The Third Edtion Is A Brief Report On Some Recent Developments In The Field And An Updated And Enlarged Supplementary Bibliography With Over 800 Items. Preface To First Edition -- Preface To Third Edition -- List Of Symbols -- Sphere Packings And Kissing Numbers -- Coverings, Lattices And Quantizers -- Codes, Designs, And Groups -- Certain Important Lattices And Their Properties -- Sphere Pakcking And Error-correcting Codes -- Laminated Lattices -- Further Connections Between Codes And Lattices -- Algebraic Constructions For Lattices -- Bounds For Codes And Sphere Packings -- Three Lectures On Exceptional Groups -- The Golay Codes And The Mathieu Groups -- A Characterization Of The Leech Lattice -- Bounds On Kissing Numbers -- Uniqueness Of Certain Spherical Codes -- On The Classification Of Integral Quadratic Forms -- Enumeration Of Unimodular Lattices -- The 24-dimensional Odd Unimodular Lattices -- Even Unimodular 24-dimensional Lattices -- Enumeration Of Extremal Self-dual Lattices -- Finding The Closest Lattice Point -- Voronoi Cells Of Lattices And Quantization Errors -- A Bound For The Covering Radius Of The Leech Lattice. The Covering Radius Of The Leech Lattice -- Twenty-three Constructions For The Leech Lattice -- The Cellular Structure Of The Leech Lattice -- Lorenzian Forms For The Leech Lattice -- The Automorphism Group Of The 26-dimensional Even Unimodular Lorenzian Lattice -- Leech Roots And Vinberg Groups -- The Moster Group And Its 196885-dimensional Space -- A Monster Lie Algebra? Bibliography. Supplemental Bibliography. By J. H. Conway, N. J. A. Sloane.