The $z$-transform is a discrete analogue to the Laplace transform, in that it maps a time domain signal into a representation in complex frequency plane.

In mathematics and its applications (notably in signal processing), the $z$-transform is a discrete transformation which maps a time domain signal into a representation in the complex frequency plane.

If $x(n)$ is a time domain signal, then the $z$-transform $X(z)$ of $x(n)$ is given formally by the power series $$X(z)=\sum_{n=-\infty}^\infty x(n)z^{-n}.$$

The $z$-transform has interesting connections to generating functions and the discrete Fourier transform.