Dynamically defined Cantor sets appear in a natural way not only in many problems of smooth dynamical systems, but also in number theory, Hamiltonian dynamics, celestial mechanics, and mathematical physics. We will review some of the known applications (dissipative and conservative Newhouse phenomena, sums of continued fractions) and discuss several new results (spectral properties of discrete Schrodinger operator with Fibonacci potential, Hausdorff dimension of a stochastic sea of the standard map, and oscillatory motions in Sitnikov problem).