Math keyboard: does it exist ? Does a keyboard specifically designed for mathematics exist (with all the common symbols and greek letters) ? 
Would you buy one and why ?
PS: Feel free to re-tag properly.
 A: One almost existed. The program I use to write TeX has several "keyboards". For example, typing Ctrl-g switches to the Greek "keyboard" and then "a" puts "\alpha" in your file; typing Ctrl-s and then "a" puts "\angle" in your file. At one time, the owners of the program were planning to produce a keyboard with the property that the symbol on the keyboard would change when you switched keyboards. So when you typed Ctrl-g, you would see $\alpha$ on the "a" key (I think they were planning to use LEDs).  It was certainly an interesting idea, but, no, I wouldn't buy such a thing --- like most people here, I touch-type, and prefer to enter everything with a sequence of normal characters.
A: There used to be a number of specialized keyboards for APL fans - like the Unicomp keyboard, which comes 'closer' than usual keyboards.  Or you could always re-program this keyboard to your heart's content!
A: I don't know the answer to your first question, but I probably would not buy one, should one exist.
(There are programmable keyboards, of course.)
A quick glance at the unicode provision for mathematical symbols (e.g., here) shows that there are too many of them to fit on a standard keyboard with 100-odd keys, even with modifiers.
This is similar, though not perhaps as severe, to the problem of trying to write in Japanese using either kanji or kana on a keyboard. (Same problem in Chinese, of course -- it's just that I've more familiarity with Japanese.) The solution there is not to have a huge keyboard with lots of symbols, but for the software to do the work.  All one would need is an input system for mathematics.  For me, although I concede it is not optimal, this is TeX.  (In fact, TeX is recognised as an input system in Emacs.)
A: Any keyboard is reprogrammable to your heart's content (assuming you're using a decent operating system).  When typing a LaTeX document, I reprogram my keyboard so that the backslash is in place of the semi-colon, the numbers and shift-numbers are swapped, and a few other optimisations.  I did this because typing LaTeX with an ordinary keyboard was causing me some pain in my fingers and doing this adjustment fixed it.
Something interesting to do is to count occurrences of characters in your documents.  For example, in my latest paper, the top four characters are:
13001 t
12385 e
11291 \
11135 o

The total number of numbers used was 766.  So making the numbers harder to type doesn't mean too much extra stretching, and putting the backslash somewhere easy to reach saves a lot of effort.

(Added in edit): A little while after this question was asked, the site http://tex.stackexchance.com came into being.  A very similar question was asked there and I gave a rather more detailed answer there than I did here.  Here's a direct link to my answer there.
(Thanks to Peter for reminding me of this and suggesting I link the two.)
A: If you are using Windows, you can use the Microsoft Keyboard Layout Creator tool to make your own layout; you will need administrator access to install it, however, and then you will have to switch to it.

As you noticed, however, no layout can cover the entire range of unicode characters.
What I have done is create an AutoHotKey script like this:
::\exists::
    SendUnicode("∃")
    return
::\forall::
    SendUnicode("∀")
    return

SendUnicode(char){
    ;send unicode encoding to clipboard because
    ;that's the only way AHK can encode Unicode.
    Transform, Clipboard, Unicode, %char%
    SendInput ^v
    return
}

I would share the whole list of bindings I've written, but it really is an inelegant, ugly kludge. However, it might do the trick for you.
If you aren't using Windows, this approach should still be available to you through other software. However, I've already used all of my link allowance. :)
A: If one is using Linux, one can set up compose sequences to have access to a large number of mathematical symbols.  If one is using Windows, one can simulate this using a freeware application called "allchars".  If one is using a Macintosh, one should be able to do something similar, but as I don't regularly use such a computer, I do not know if a third-party application is necessary.
A: Does an APL keyboard (e.g., http://www.aplusdev.org/keyboard.html) qualify?
A: All keyboards are more or less the same, the only difference is in the symbols printed on the keys.
Hence the question is really about math keyboard layouts.
My keyboard layout has a lot of math symbols and use them every time I type math.
Some examples (typed using my keyboard layout!): α β Γ Δ ψ Ψ (all greek letters are included),
≤ ≥ ⊗ → ∈ ∞ ↦ ≠ 〈 〉 ⊂ ⊃ ⊆ ⊇ ≅ ≃
TeX has about 230 math symbols (including Greek letters).
You can easily encode them all, but my keyboard layout has only about 50 most commonly used symbols
and 24*2=48 Greek letters.
And note that you can easily setup TeX to recognize Unicode math symbols in UTF-8 encoding.
Incidentally, I also have commonly used punctuation marks that are absent from ASCII: “ ” ‘ ’ – — … 
A: A German group developed a new keyboard layout, called neo (similar to the Dvorak layout), which is optimized for German words, but also for English.
This layout has six layers; layer 5 is for Greek chars and layer 6 is for math symbols.
You can check their website: http://www.neo-layout.org (in German; you can also consider the English translation of the site by Google).
A: I actually have an onscreen keyboard that has these layouts.
Here you can find a bunch of layouts, the one called "Special" has Greek, and as many math symbols as I could stuff in, using shift, ctrl, and combinations.
∩∈⊂⊆∼≃≅≈≊× =
∃∄ℯℜ √∐ℹℴ℘∣∥≡
∀⅀∂ℱℏℌ∫∮ℓ°∞
ℤ∠∁∇ℬℕℳ<> 
≌≐≝⋍
ℚ ℰℛ∜ ℑ∅ℿ
ℵℋ∭∰∱∲
⊕⊗⊢⊣⊤⊥⊨⊩⊬
∪∉⊄⊈≁≄≇≉≉≠
ℇ∛ ℐ∅ℙ∤∤∦≢
∆ℾℍ∬∯ℒ∙∝
 ≥≰≱≮≯
Can use it on the web.
A: If you are using the TeX-Editor WinEdt you can do the following:
Go to Options->Settings->Translations->Keyboard.
You you can completely customize your keyboard. 
For example I use ";" -> "$" in order to type the dollar symbol more conveniently.
A: Optimus Maximus [1] can become just what you need.
[1] http://www.artlebedev.com/everything/optimus/
A: Might be a nice touchscreen app, even for a fairly small touchscreen device like an Android phone.  Most text typing would be on the computer's normal keyboard, but symbols could be entered on a "virtual" touchscreen keyboard.  The software would be set up so that you could type both keyboards simultaneously without having to mouse around and so forth.  Hmm, maybe I'll try implementing this.
A: For the record, I do NOT find this question off-topic.
Math is moving away from the pen and paper (or chalk and chalkboard) days.  Computers are everywhere, and, unlike days of old when computers were notoriously BAD at math, modern languages, such as Python, have built in representations of fractions, complex numbers, positive and negative infinity, etc. 
Matrices and vectors, for example, are used in everything from the latest shoot-em-up to bank transactions, to simulation software and drawing programs.
It is about time that we had a real solution for entering and working with real equations on a computer, rather than having to slog it out with --> area = (PI * (r^2)).
Would I buy a keyboard that had all the standard mathematical symbols built into it?
Absolutely.
And while, yes, I am aware that I can remap a standard keyboard, and while yes, I touch type... some of these symbols would be used often enough to want on the keyboard, yet rarely enough to have no idea where you put them.  (Where the %&@# did I put phi?  CTRL-P? Bah, PI... Alt P?  Rho?  Whats that doing there?)... a quick look down on the proper keyboard could solve this easily.
And yes... I came to this board after doing a Google search for just such a beast... if this question had not been asked here, I never would have wandered to your site.
Me
A: I don't care that I can't actually touch-type all those keys; as long as all the mathematical keys, Greek letters and European accents are somewhere on my desk, that's good enough. I want to be able to press it, not fiddle around in software to insert my characters.
I think it'd be best as an extra keyboard just with these characters, rather than a replacement for QWERTY. Is it possible to have 2 keyboards like this? I'd definitely buy one
