Informally speaking, fractals are sets of non-integer dimension. The
standard Cantor set is a simplest example of a fractal set. An important
characteristic of a fractal set is its fractal (or Hausdorff) dimension. We
will give two examples of recent results (oscillatory motions in the three
body problem and spectrum of a discrete Schrodinger operator with Fibonacci
potential) where fractals appear in a natural way, and their Hausdorff
dimension can be estimated.