Quantum uncertainty relations for infinite-dimensional Hilbert space are recalled. Apart of the standard, 1927 approach of Heisenberg-Kennard, based on the product of variances of the probability distribution in two different representations, we advocate an alternative 1975 approach of Bialynicki-Birula and Mycielski, in which sum of two entropies characterizing both distributions is investigated. In the case of a finite-dimensional Hilbert space we present the 1983 bounds of Deutsch and 1988 of Maassen and Uffink and discuss possibilities to improve them. In particular, a generic case of two orthogonal measurements performed in two bases related by a random unitary matrix is analyzed. Complementary part of the talk is focused on properties of quantum wave functions - solutions of the Schroedinger equation for model systems. If a quantum system is described in an infinite Hilbert space, such wave functions can display genuinely fractal properties, which in physical systems can be observed only up to a certain accuracy. However, such nowhere differentiable functions are by no mean exceptional, as they form a prevalent set in the space of all continuous functions.