Chiral polyhedra in ordinary euclidean space E~3 are nearly regular polyhedra; their geometric symmetry groups have two orbits on the flags, such that adjacent flags are in distinct orbits. This paper completely enumerates the discrete infinite chiral polyhedra in E~3 with finite skew faces and finite skew vertex-figures. There are several families of such polyhedra of types {4,6}, {6,4} and {6,6}. Their geometry and combinatorics are discussed in detail. It is also proved that a chiral polyhedron in E~3 cannot be finite. Part II of the paper will complete the classification of all chiral polyhedra in E~3. All chiral polyhedra not described in Part I have infinite, helical faces and again occur in families. So, in effect, Part I enumerates all chiral polyhedra in E~3 with finite faces.
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