  In the Disjoint Paths problem, the input consists of an $n$-vertex graph $G$
and a collection of $k$ vertex pairs, $\{(s_i,t_i)\}_{i=1}^k$, and the
objective is to determine whether there exists a collection $\{P_i\}_{i=1}^k$
of $k$ pairwise vertex-disjoint paths in $G$ where the end-vertices of $P_i$
are $s_i$ and $t_i$. This problem was shown to admit an $f(k)n^3$-time
algorithm by Robertson and Seymour (Graph Minors XIII, The Disjoint Paths
Problem, JCTB). In modern terminology, this means that Disjoint Paths is fixed
parameter tractable (FPT) with respect to $k$. Remarkably, the above algorithm
for Disjoint Paths is a cornerstone of the entire Graph Minors Theory, and
conceptually vital to the $g(k)n^3$-time algorithm for Minor Testing (given two
undirected graphs, $G$ and $H$ on $n$ and $k$ vertices, respectively, determine
whether $G$ contains $H$ as a minor).
  In this semi-survey, we will first give an exposition of the Graph Minors
Theory with emphasis on efficiency from the viewpoint of Parameterized
Complexity. Secondly, we will review the state of the art with respect to the
Disjoint Paths and Planar Disjoint Paths problems. Lastly, we will discuss the
main ideas behind a new algorithm that combines treewidth reduction and an
algebraic approach to solve Planar Disjoint Paths in time
$2^{k^{O(1)}}n^{O(1)}$ (for undirected graphs).
