If I save a notebook, I only save what I see. That is, if I close Mathematica and reopen the notebook later, all previous calculations are lost, except for those things I've output. Now I know that you can save single definitions (and their dependencies) with Save, however is there a way to save the whole kernel state (preferably including internally cached data, e.g. from FullSimplify) so that when I reload both notebook and kernel state, I can continue to work exactly where I left off?

If there's no ready solution for it:

From what I understand, the explicit definitions are stored in UpValues, DownValues and OwnValues; also Options and Attributes affect evaluation, and therefore would have to be saved. However, those all expect an argument specifying a symbol to give information about. Is there any way to get a complete set of them? And would saving those actually suffice, or is there something else needed, too? Also, is there some way to explicitly get at internal cached information (it doesn't need to be in an understandable format, just being able to save and reload it would be sufficient)?