  The theory of Non-Relativistic Quantum Mechanics was created (or discovered)
back in the 1920's mainly by Schr\"odinger and Heisenberg, but it is fair
enough to say that a more modern and unified approach to the subject was
introduced by Dirac and Jordan with their (intrinsic) Transformation Theory. In
his famous text book on quantum mechanics [1], Dirac introduced his well-known
bra and ket notation and a view that even Einstein (who was, as well known,
very critical towards the general quantum physical world-view) considered the
most elegant presentation of the theory at that time[2]. One characteristic of
this formulation is that the observables of position and momentum are truly
treated equally so that an intrinsic phase-space approach seems a natural
course to be taken. In fact, we may distinguish at least two different quantum
mechanical approaches to the structure of the quantum phase space: The
Weyl-Wigner (WW) formalism and the advent of the theory of Coherent States
(CS). The Weyl-Wigner formalism has had many applications ranging from the
discussion of the Classical/Quantum Mechanical transition and quantum chaos to
signal analysis[3,4]. The Coherent State formalism had a profound impact on
Quantum Optics and during the course of time has found applications in diverse
areas such as geometric quantization, wavelet and harmonic analysis [5]. In
this chapter we present a compact review of these formalisms (with also a more
intrinsic and coordinate independent notation) towards some non-standard and
up-to-date applications such as modular variables and weak values.
