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A fully consistent classical relativistic electrodynamics with spinless pointcharges is constructed. The classical evolution of the electromagnetic fieldsis governed by the nonlinear Maxwell--Born--Infeld field equations, theclassical evolution of the point charges by a many-body Hamilton--Jacobi law ofmotion. The Pauli principle for bosons can be incorporated in the classicalHamilton--Jacobi formalism. The Cauchy problem is explained and illustratedwith examples. The question of charge-free field solitons is addressed also andit is shown that if they exist, their peak field strengths must be enormous.The value The value of Born's constant is shown to be a subtle open issue.Comment: Minor corrections at galley stage incorporated. 66p; to appear in JSP  vol. 116, issue dedicated to Elliott H. Lieb on his 70th birthday. Part II is  math-ph/031103

Electromagnetic Field Theory Without Divergence Problems 1. The Born Legacy