Lists of mathematical problems

Hilbert's problems

He's a bit overly obsessed with polynomials for the taste of modern maths, but it's still fun.

Ciro Santilli would like to fully understand the statements and motivations of each the problems!

Easy to understand the motivation:

Navier-Stokes existence and smoothness is basically the only problem that is really easy to understand the statement and motivation :-)
p versus NP problem

Hard to understand the motivation!

Riemann hypothesis: a bunch of results on prime numbers, and therefore possible applications to cryptography
Of course, everything of interest has already been proved conditionally on it, and the likely "true" result will in itself not have any immediate applications.
As is often the case, the only usefulness would be possible new ideas from the proof technique, and people being more willing to prove stuff based on it without the risk of the hypothesis being false.
Yang-Mills existence and mass gap: this one has to do with findind/proving the existence of a more decent formalization of quantum field theory that does not resort to tricks like perturbation theory and effective field theory with a random cutoff value
This is important because the best theory of light and electrons (and therefore chemistry and material science) that we have today, quantum electrodynamics, is a quantum field theory.