Quantum fluctuations generate in three-dimensional gauge theories not only radiative corrections to the Chern-Simons coupling but also non-analytic terms in the effective action. We review the role of those terms in gauge theories with massless fermions and Chern-Simons theories. The explicit form of non-analytic terms turns out to be dependent on the regularization scheme and in consequence the very existence of phenomena like parity and framing anomalies becomes regularization dependent. In particular we find regularization regimes where both anomalies are absent. Due to the presence of non-analytic terms the effective action becomes not only discontinuous but also singular for some background gauge fields which include sphalerons. The appearence of this type of singularities is linked to the existence of nodal configurations in physical states and tunneling suppression at some classical field configurations. In the topological field theory the number of physical states may also become regularization dependent. Another consequence of the peculiar behaviour of three-dimensional theories under parity odd regularizations is the existence of a simple mechanism of generation of a mass gap in pure Yang-Mills theory by a suitable choice of regularization scheme. The generic value of this mass does agree with the values obtained in Hamiltonian and numerical analysis. Finally, the existence of different regularization regimes unveils the difficulties of establishing a Zamolodchikov c-theorem for three-dimensional field theories in terms of the induced gravitational Chern-Simons couplings.