When should I really use noexcept?

The noexcept keyword can be appropriately applied to many function signatures, but I am unsure as to when I should consider using it in practice. Based on what I have read so far, the last-minute addition of noexcept seems to address some important issues that arise when move constructors throw. However, I am still unable to provide satisfactory answers to some practical questions that led me to read more about noexcept in the first place.

1. There are many examples of functions that I know will never throw, but for which the compiler cannot determine so on its own. Should I append noexcept to the function declaration in all such cases?

   Having to think about whether or not I need to append noexcept after every function declaration would greatly reduce programmer productivity (and frankly, would be a pain in the ass). For which situations should I be more careful about the use of noexcept, and for which situations can I get away with the implied noexcept(false)?

2. When can I realistically expect to observe a performance improvement after using noexcept? In particular, give an example of code for which a C++ compiler is able to generate better machine code after the addition of noexcept.

   Personally, I care about noexcept because of the increased freedom provided to the compiler to safely apply certain kinds of optimizations. Do modern compilers take advantage of noexcept in this way? If not, can I expect some of them to do so in the near future?