This is a survey paper on the general theory of fractals and fractal measures containing several proofs and exercises. It has the style of Lecture Notes and is suitable for a short introduction on this subject. The paper is concentrated on questions which have not been completely worked out in the main reference books of this subject as are \textit{K. J. Falconer}: ``The geometry of fractals sets'' (1985; Zbl 0587.28004) and ``Fractal geometry: mathematical foundation and applications'' (1990; Zbl 0689.28003), and \textit{G. A. Edgar}: ``Measure, topology, and fractal geometry (1990; Zbl 0727.28003). In section 1 the hierarchical structure of Cantor-type sets and Cantor measures is described by trees and Markovian processes. In section 2 fractals are described by iterated function schemes and random algorithms. In section 3 and 4 several classes of invariant measures and their dimension are examined. Finally, section 4 contains an overview of the Sierpiński gasket.