This volume is intended to collect important research results to the lectures and discussions which took Place in Rome, at the INdAM Workshop on Different Notions of Regularity for Functions of Quaternionic Variables in September 2010. This volume will collect recent and new results, which are connected to the topic covered during the workshop. The work aims at bringing together international leading specialists in the field of Quaternionic and Clifford Analysis, as well as young researchers interested in the subject, with the idea of presenting and discussing recent results, analyzing new trends and techniques in the area and, in general, of promoting scientific collaboration. Particular attention is paid to the presentation of different notions of regularity for functions of hypercomplex variables, and to the study of the main features of the theories that they originate.

C. Bisi, C. Stoppato: Regular vs. classical Mobius transformations of the quaternionic unit ball.- F. Brackx, H. De Bie, Hennie De Schepper: Distributional Boundary Values of Harmonic Potentials in Euclidean Half-space as Fundamental Solutions of Convolution Operators in Clifford Analysis.- F. Colombo, J.O. Gonzalez-Cervantes, M.E. Luna-Elizarraras, I. Sabadini, M. Shapiro: On two approaches to the Bergman theory for slice regular functions.- C. Della Rocchetta, G. Gentili, G. Sarfatti: A Bloch-Landau Theorem for slice regular functions.- M. Ku, U. Kahler, P. Cerejeiras: Dirichlet-type problems for the iterated Dirac operator on the unit ball in Clifford analysis.- A. Perotti: Fueter regularity and slice regularity: meeting points for two function theories.- D.C. Struppa: A note on analytic functionals on the complex light cone.- M.B. Vajiac: The S-spectrum for some classes of matrices.- F. Vlacci: Regular Composition for SliceRegular Functions of Quaternionic Variable.