In this paper we develop the calculus of pseudo-differential operators
corresponding to the quantizations of the form $$
Au(x)=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}e^{i(x-y)\cdot\xi}\sigma(x+\tau(y-x),\xi)u(y)dyd\xi,
$$ where $\tau:\mathbb{R}^n\to\mathbb{R}^n$ is a general function. In
particular, for the linear choices $\tau(x)=0$, $\tau(x)=x$, and
$\tau(x)=\frac{x}{2}$ this covers the well-known Kohn-Nirenberg,
anti-Kohn-Nirenberg, and Weyl quantizations, respectively. Quantizations of
such type appear naturally in the analysis on nilpotent Lie groups for
polynomial functions $\tau$ and here we investigate the corresponding calculus
in the model case of $\mathbb{R}^n$. We also give examples of nonlinear $\tau$
appearing on the polarised and non-polarised Heisenberg groups, inspired by the
recent joint work with Marius Mantoiu.