Jacobian conjecture: Let $k$ be an algebraically closed field of characteristics zero, $n\geq 2$ and $\phi: k^n\to k^n$ a (regular) endomorphism of $k^n$ with constant Jacobian (the determinant of the Jacobian matrix, which is in this polynomial case algebraically defined). Then $\phi$ is a regular automorphism, i.e. has a polynomially defined inverse.
The conjecture is open still stated by Keller in 1939. There were many failed attempts to prove the Jacobian conjecture, especially for $n = 2$; there are also some reductions to special cases. For example, it is known that the Jacobian conjecture holds iff it holds for $\phi$ a polynomial map of degree 3. The Jacobian conjecture is also known to hold at least for those $\phi$ which have a rational inverse.
A. van den Essen, Jacobian conjecture, Springer Online Encyclopedia of Mathematics
A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, pdf, Algèbre non commutative, groupes quantiques et invariants (Reims, 1995), 55–81, Sémin. Congr., 2, Soc. Math. France, Paris, 1997.
Arno van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, 190. Birkhäuser Verlag, Basel, 2000. xviii+329 pp. ISBN: 3-7643-6350-9
The Jacobian conjecture is also equivalent to the Dixmier conjecture: every endomorphism of the $r$-th Weyl algebra $A_{r,k}$ over $k$ is an automorphism for all $r$. This is a statement of the article
which does contain an error in the proof, which has been later amended by others. It is actually known that each endomorphism of the $r$-th Weyl algebra is injective, and not known wheather it is surjective. A shorter algebraic proof is given in
A claim of a pretty short proof of the general case of Jacobian conjecture (no essential error found so far?) is in
There is an interesting blog discussion, from the point of view of algebraic geometry: