  In this article, we study isomorphisms between complements of irreducible
curves in the projective plane $\mathbb{P}^2$, over an arbitrary algebraically
closed field. Of particular interest are rational unicuspidal curves. We prove
that if there exists a line that intersects a unicuspidal curve $C \subset
\mathbb{P}^2$ only in its singular point, then any other curve whose complement
is isomorphic to $\mathbb{P}^2 \setminus C$ must be projectively equivalent to
$C$. This generalizes a result of H. Yoshihara who proved this result over the
complex numbers. Moreover, we study properties of multiplicity sequences of
irreducible curves that imply that any isomorphism between the complements of
these curves extends to an automorphism of $\mathbb{P}^2$. Using these results,
we show that two irreducible curves of degree $\leq 7$ have isomorphic
complements if and only if they are projectively equivalent. Finally, we
describe new examples of irreducible projectively non-equivalent curves of
degree $8$ that have isomorphic complements.
