Is there a calculus of non-commutative variables?

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I know the integral calculus for commuting variables (which is just normal calculus).

And there is an Grassman integral calculus for anti-commuting variables. (Variables where $\theta_1 \theta_2 = - \theta_2\theta_1$ ).

Is there a calculus for general variables that in general satisfy:

$$x y\neq y x$$

Perhaps it would be a calculus of matrices where $x$ and $y$ are general matrices?