  In 1968, N.\,V.~Efimov proved 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 $a(x)>0$ 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 $|\det f'(x)|\geqslant
a(x)|\operatorname{curl}f(x)|+a^2(x)$ 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
$f(\mathbb{R}^2)$ homeomorphically.} Here $\operatorname{curl}f(x)$ 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 global inverse functions, as well as in the study of the
Jacobian Conjecture and the global asymptotic stability of dynamical systems.

Differential Geometry;Algebraic Geometry;Classical Analysis and ODEs;Dynamical Systems