By examining the complex relationship between pi and the AGM, or Arithmetic-Geometric Mean, this book presents research revealing the interplay between classical analysis and modern computation and complexity theory. Two intimately-interwoven strands run through the book: the AGM iteration of Gauss, Lagrange, and Legendre, and the calculation of pi. Also included in the discussion are Rogers-Ramanujan identities, algebraic series for pi, results on sums of two and four squares, the transcendence of pi and Madelun's constant, lattice sums, and elliptical invariants.