  Let $\Theta$ be a variety of algebras. In every $\Theta$ and every algebra
$H$ from $\Theta$ one can consider algebraic geometry in $\Theta$ over $H$. We
consider also a special categorical invariant $K_\Theta (H)$ of this geometry.
The classical algebraic geometry deals with the variety $\Theta=Com-P$ of all
associative and commutative algebras over the ground field of constants $P$. An
algebra $H$ in this setting is an extension of the ground field $P$. Geometry
in groups is related to varieties $\Grp$ and $\Grp-G$, where $G$ is a group of
constants. The case $\Grp -F$ where $F$ is a free group, is related to Tarski's
problems devoted to logic of a free group. he described general insight on
algebraic geometry in different varieties of algebras inspires some new
problems in algebra and algebraic geometry. The problems of such kind
determine, to a great extent, the content of universal algebraic geometry. We
start with the short overview of main definitions and results and then consider
the list of unsolved problems.
