Stability of the Solar System 
Is the Solar System stable? 

You can see this Wikipedia page.
In May 2015 I was at the conference of Cedric Villani at Sharif university of technology with this title: "Of planets, stars and eternity (stabilization and long-time behavior in classical celestial mechanics)"
, at the end of this conference one of the students asked him this question and he laughed strangely(!) with no convincing answer!
Edit: The purpose of "long-time" is timescale more than Lyapunov time, hence billions of years.
 A: in a conference in Paris, Jacques Féjoz said (and i quote from memory) that the big planets seem to be stable, while the small ones chaotic. if i remember well, it was based on numerical evidence, intuition and the known results on the stability of the planar many-body problem...
A: The work of Wisdom is relevant (for example, see http://rspa.royalsocietypublishing.org/content/413/1844/109.short) but not conclusive.  Numerical work suggests overall stability over very long time periods.
A: Due to chaotic behaviour of the Solar System, it is not possible to precisely predict the evolution of the Solar System over 5 Gyr and the question of its long-term stability can only be answered in a statistical sense. For example, in
http://www.nature.com/nature/journal/v459/n7248/full/nature08096.html (Existence of collisional trajectories of Mercury, Mars and Venus with the Earth, by 
J. Laskar and M. Gastineau) 2501 orbits with different initial conditions all consistent with our present knowledge of the parameters of the Solar System were
traced out in computer simulations up to 5 Gyr. The main finding of the paper is that one percent of the solutions lead to a large enough increase in Mercury's eccentricity to allow its collisions with Venus or the Sun.
Probably the most surprising result of the paper (see also http://arxiv.org/abs/1209.5996) is that in a pure Newtonian world the probability of collisions within 5 Gyr grows to 60 percent and therefore general relativity is crucial for long-term stability of the inner solar system.
Many questions remain, however, about reliability of the present day consensus that the odds for the catastrophic destabilization of the inner planets are in the order of a few percent. I do not know if the effects of galactic tidal perturbations or possible perturbations from passing stars are taken into account. Also different numerical algorithms lead to statistically different results (see, for example, http://arxiv.org/abs/1506.07602).
Some interesting historical background of solar system stability studies can be found in http://arxiv.org/abs/1411.4930 (Michel Henon and the Stability of the Solar System, by Jacques Laskar).
A: This paper (Batyrin and Laughlin, 2008) seems to indicate that we are doomed.
