We give a new classification of tilings of the 2-dimensional sphere by
congruent triangles accompanied with a complete proof. This accomplishes the old
classification by Davies, who only gave an outline of the proof, regrettably with some
redundant tilings. We clarify Davies’ obscure points, give a complete list, and show
that there exist ten sporadic and also ten series of such tilings, including some unfamiliar
twisted ones. We also give their figures, development maps in a way easy to understand
their mutual relations. In Appendix, we give curious examples of tilings on noncompact
spaces of constant positive curvature with boundary possessing a special 5-
valent vertex that never appear in the tiling of the usual sphere.