In this talk, I will give a brief introduction to 
"parameterized complexity", one of the latest proposals 
on how to deal with combinatorically hard problems. A 
parameterized problem is called fixed parameter tractable 
if it admits a solving algorithm whose running time on input 
instance $(I,k)$ is $f(k) \cdot |I|^\alpha$, where $f$ is 
an arbitrary function depending only on the problem 
parameter $k$.
 I want to focus on obtaining a new qualitative behaviour of 
the exponential function $f$ by presenting different techniques 
for designing linear time algorithms with $f(k)=c^{\sqrt{k}}$
for various problems on planar graphs (including Vertex Cover, 
Independent Set, Dominating Set, or Face Cover).
One of the methods presented in the talk is based on tree 
decompositions. 
 Besides, I will report on experimental results  with respect to our
corresponding software package that implements the discussed algorithms
using the LEDA library. Compared to our worst case analysis, the 
algorithms behave surprisingly well in practice and allow to attack 
problem instance sizes not considered feasible before.