This is a special issue of the journal devoted to the jubilee of the outstanding Ukrainian and Russian mathematician, one of the founder of our journal Professor Rostislav Grigorchuk, who was 50 years of age on February 23, 2003. R. Grigorchuk was born in Ternopol oblast, Ukraine. He left the Ukrainian school no.23 in Chernivtsi, Ukraine. In 1975, after graduating Moscow Lomonosov State University (faculty of Mechanics and Mathematics), he entered the post-graduate course in the same university His scientific supervisor was the well-known specialist in ergodic theory and dynamic systems Professor A. M. Stepin.

In 1978, R. Grigorchuk defended his Ph.D. thesis: ``Banach means on homogeneous spaces and random walks'' in Moscow Lomonosov State University. The degree of Doctor of Sciences he received in 1985 in Steklov Institute of Mathematics for his thesis ``Growth functions of finitely generated groups and its applications''.

His teaching work R.I. Grigorchuk began in Department of Mathematics of Moscow State University of Transportation; at first as Assistant Professor, and then from 1992 till 1995 as Chair of Department. Such known mathematicians as Yurii Kuzmin, Zoya Lipkina, Leonid Sadovsky and Elena Ventzel worked at this department worked at that time. In 1955 R.I.Grigorchuk passed to the Steklov Institute of Mathematics where he is working till now as the leading scientific researcher. He also is Professor of the Chair of Dynamical Systems of Moscow State Lomonosov University. This chair was organized with his active assistance.

Professor Grigorchuk is a mathematician of a wide range of interests and the results he obtained belong to various areas of Mathematics: group theory, as a whole, and such its directions as asymptotic group theory and geometric group theory, branched groups and groups generated by finite automata, problems of Burnside's type, growth of groups (around Milnor's problem), dynamic systems. In particular, ergodic theorems for non-commutative groups of transformations. Theory of invariant means (amenable groups), random walks on groups and graphs. Spectral theory of discrete of Laplacian, bounded cohomologies and $L^{2}$-invariants. Low dimensional topology and discrete Mathematics.

In the paper ``On Burnside's problem on periodic groups'' (Functional Anal. Appl., v.14, no.1, 1980, 53-54) R.I.Grigorchuk constructed his first example of infinite finitely generated periodic group. This group is known as the first Grigorchuk group and it represents the simplest example of the so-called groups of Burnside type. The first Grigorchuk group has had different applications in various branches of mathematics and can be regarded as the pearl in the collection of nice and useful algebraic objects. R.I. Grigorchuk solved a series of important problems in asymptotic and geometric group theory, in particular, he solved the famous Milnor's problem about groups of intermediate growth. It is impossible to imagine modern group theory without deep theorems and nice constructions of R.I. Grigorchuk. Outstanding achievements in science of R.I.Grigorchuk were highly rated by mathematical community, he was repeatedly awarded to the prizes for the best scientific results in Russia (1999, 2000 and 2001), and to the honorary title ``Outstanding scientist of Russia'' in 1996 and 1999. Many times he has been invited by world mathematical centers for giving series of lectures in rapidly developing areas of Mathematics.

Professor R.I. Grigorchuk is a well-known organizer of mathematical researches, in particular, in Russia and in Ukraine. With his active assistance, International Algebraic conferences in Ukraine in 1997, 1999, 2001 and 2003 were organized. He was a participant of the Ukrainian Mathematical Congress 2001.

Practically every year R.I. Grigorchuk carries out winter school on geometric group theory in Kiev Taras Shevchenko University. In collaboration with V.I. Sushchansky he developed a new direction of scientific investigations in Kiev University, namely, geometric and asymptotic group theory.

We joint to all warm wishes and congratulations which were given to Prof. Grigorchuk on the occasion of his 50th birthday and also wish him good health and long years of further activity as a scientists and teacher. As it is usually said in Ukraine ``Ç ðîñè ³ âîäè Âàì, äîðîãèé ïàíå Ðîñòèñëàâå!''