Definition 37.20.1. Let $f : X \to Y$ be a morphism of schemes. Assume that all the fibres $X_ y$ are locally Noetherian schemes.
Let $x \in X$, and $y = f(x)$. We say that $f$ is regular at $x$ if $f$ is flat at $x$, and the scheme $X_ y$ is geometrically regular at $x$ over $\kappa (y)$ (see Varieties, Definition 33.12.1).
We say $f$ is a regular morphism if $f$ is regular at every point of $X$.