  We develop geometry of affine algebraic varieties in $K^{n}$ over Henselian
rank one valued fields $K$ of equicharacteristic zero. Several results are
provided including: the projection $K^{n} \times \mathbb{P}^{m}(K) \to K^{n}$
and blow-ups of the $K$-rational points of smooth $K$-varieties are definably
closed maps, a descent property for blow-ups, curve selection for definable
sets, a general version of the \L{}ojasiewicz inequality for continuous
definable functions on subsets locally closed in the $K$-topology and extending
continuous hereditarily rational functions, established for the real and
$p$-adic varieties in our joint paper with J. Koll\'ar. The descent property
enables application of resolution of singularities and transformation to a
normal crossing by blowing up in much the same way as over the locally compact
ground field. Our approach relies on quantifier elimination due to Pas and a
concept of fiber shrinking for definable sets, which is a relaxed version of
curve selection. The last three sections are devoted to the theory of regulous
functions and sets over such valued fields. Regulous geometry over the real
ground field $\mathbb{R}$ was developed by Fichou--Huisman--Mangolte--Monnier.
The main results here are regulous versions of Nullstellensatz and Cartan's
Theorems A and B.
