  This thesis consists of Chapters 1 and 2. The main results are contained in
the two preprints and two published papers, listed below. Chapter 1 deals with
conformal invariants in the euclidean space Rn; n >= 2; and their
interrelation. In particular, conformally invariant metrics and balls of the
respective metric spaces are studied. Another theme in Chapter 1 is the study
of quasiconformal maps with identity boundary values in two diferent cases, the
unit ball and the whole space minus two points. These results are based on the
two preprints: R. Klen, V. Manojlovic and M. Vuorinen: Distortion of two point
normalized quasiconformal mappings, arXiv:0808.1219[math.CV], 13 pp., V.
Manojlovic and M. Vuorinen: On quasiconformal maps with identity boundary
values, arXiv:0807.4418[math.CV], 16 pp. Chapter 2 deals with harmonic
quasiregular maps. Topics studied are: Preservation of modulus of continuity,
in particular Lipschitz continuity, from the boundary to the interior of domain
in case of harmonic quasiregular maps and quasiisometry property of harmonic
quasiconformal maps. Chapter 2 is based mainly on the two published papers: M.
Arsenovic, V. Kojic and M. Mateljevic: On Lipschitz continuity of harmonic
quasiregular maps on the unit ball in Rn., Ann. Acad. Sci. Fenn. Math. 33
(2008), no. 1, 315-318. V. Kojic and M. Pavlovic: Subharmonicity of jfjp for
quasiregular harmonic functions, with applications, J. Math. Anal. Appl. 342
(2008) 742-746
