  In the year 1968 N.V. Efimov has proven the following remarkable theorem:
  \textit{Let $f:\mathbb R^2\to\mathbb R^2\in C^1$ be such that $\det f'(x)<0$
for all $x\in\mathbb R^2$ and let there exist a function $a=a(x)>0$ and
constants $C_1\geqslant 0$, $C_2\geqslant 0$ such that the inequalities
$|1/a(x)-1/a(y)|\leqslant C_1 |x-y|+C_2$ and $|\det f'(x)|\geqslant a(x)|{\rm
curl\,}f(x)|+a^2(x)$ hold true for all $x, y\in\mathbb R^2$. Then $f(\mathbb
R^2)$ is a convex domain and $f$ maps $\mathbb R^2$ onto $f(\mathbb R^2)$
homeomorhically.}
  Here ${\rm curl\,}f(x)$ stands for the curl of $f$ at $x\in\mathbb R^2$.
  This article is an overview of analogues of this theorem, its generalizations
and applications in the theory of surfaces, theory of functions, as well as in
the study of the Jacobian conjecture and global asymptotic stability of
dynamical systems.
