  We announce here a number of results concerning representation theory of the
algebra $R=k<x,y>/ (xy-yx-y^2)$, known as Jordan plane (or Jordan algebra).
  We consider the question on 'classification' of finite-dimensional modules
over the Jordan algebra. Complete description of irreducible components of the
representation variety $mod (R,n)$, which we call a Jordan variety' is given
for any dimension $n$. It is obtained on the basis of the stratification of
this variety related to the Jordan normal form of $Y$. Any irreducible
component of the representation variety contains only one stratum related to a
certain partition of $n$ and is the closure of this stratum. The number of
irreducible components therefore is equal to the number of partitions of $n$.
  As a preparation for the above result we describe the complete set of
pairwise non-isomorphic irreducible modules $S_{a}$ over the Jordan algebra,
and the rule how they could be glued to indecomposables. Namely, we show that
${\rm Ext}^1(S_{a},S_{b})=0$, if $a \neq b $.
  We study then properties of the image algebras in the endomorphism ring.
Particularly, images of representations from the most important stratum,
corresponding to the full Jordan block $Y$. This stratum turns out to be the
only building block for the analogue of the Krull-Remark-Schmidt decomposition
theorem on the level of irreducible components. Along this line we establish an
analogue of the Gerstenhaber--Taussky--Motzkin theorem on the dimension of
algebras generated by two commuting matrices. Another fact concerns with the
tame-wild question for those image algebras. We show that all image algebras of
$n$-dimensional representations are tame for $n \leq 4$ and wild for $n \geq
5$.
