We present several results on relations between the Kolmogorov
 complexity of infinite strings and measures of information content
 (dimensions) known from dimension theory, information theory or fractal
 geometry.
   
 Special emphasis is laid on bounds on the complexity of strings in
self-similar sets in Cantor space. Here a set is called self-similar 
iff it is the maximum solution of an equation of the form F = WF
where W is a set of finite strings of length > 0 and F is the subset 
of Cantor space defined by this equation.

It turns out that the Hausdorff dimension of these sets can be
expressed in terms of the entropy of the language W, and that 
the Kolmogorov Complexity (or constructive dimension) 
of infinite strings in F can be estimated via Hausdorff dimension
or entropy of languages.

For sets in Cantor space definable by finite automata 
(regular \omega-languages) even tighter bounds of 
Kolmogorov Complexity via Hausdorff dimension
or entropy of languages are obtained. Here results 
generalizing P. Martin-L"of's on the bounds
for Kolmogorov complexity and oscillation of the complexity
in an infinite string are obtained.