Abstract: 
The Kepler equations are nonlinear equations consisting the essential part of the socalled Keplerian motion, the fundamental solution of the twobody problem. It is easy to obtain the size, shape, and orientation of the orbit from the given orbital elements. However, we must solve appropriate Kepler equations to determine the position on the orbit. Depending on the type of orbit, there is a variety of Kepler equations such as the elliptic, parabolic, hyperbolic, and so on. Except the parabolic case, no analytical solutions exist so that these equations must be solved numerically. Their precise and fast solution had been quite difficult problems from the days of Kepler, Newton, and Halley. Once the author developed new solutions of all Kepler equations (Fukushima, 1996, 1997a, 1997b, 1997c, 1998, 1999). They are sufficiently precise and so fast that they run 213 times faster than the existing methods of solution. This talk shall describe the essential part of the new methods. The FORTRAN software packages of the new solutions such as xekep2.txt are freely obtained from https://www.researchgate.net/profile/ToshioFukushima/
