I will discuss divisibility and wild kernels in algebraic K-theory of number fields $F$ and present basic results concerning the divisible elements in K-groups. Without appealing to the Quillen-Lichtenbaum conjecture one can prove that the group of divisible elements is isomorphic to the corresponding group of \' etale divisible elements. One can apply this result for the proof of the $lim^1$ analogue of the Quillen-Lichtenbaum conjecture. One can also apply it to investigate: the imbedding obstructions in homology of $GL,$ the splitting obstructions for the Quillen localization sequence, the order of the group of divisible elements via special values of $\zeta_{F}(s).$ I will discuss the relation of divisible elements with Kummer-Vandiver and Iwasawa conjectures. I will also present recent results, joint with Cristian Popescu, concerning Brummer-Stark conjecture and Galois equivariant Stickelberger splitting map in Quillen localization sequence. The Stickelberger splitting map is a basic tool to investigate the structure of the group of divisible elements.