The Weyl–Wigner quantization provides a natural and consistent recipe for travelling between the classical and quantum worlds. Nevertheless, its properties are not obvious even in the most natural examples. In this talk, I will thoroughly analyze the quantization of Gaussian functions on the phase space. I will speak about its various properties and some semigroups stemming from it. I will also briefly describe its application to the proof of the renowned Calderon–Vailancourt theorem of L2–boundedness of quantization in the Weyl–Wigner case.