The main scientific fields are Information Theory, Prediction, Complexity of Algorithms,  
Cryptography and Mathematical Biology.

The main results are the following.

INFORMATION THEORY and PREDICTION
1979---
I proved the result equating the minimax redundancy for a universal code to a channel capacity. It's very important for source coding theory because a channel capacity is the main concept of Information Theory. This result was independently discovered by R. Gallager ,but was not published.(see my note in : IEEE Trans.Inform Theory,1981,v.27,n 6,pp.780-781,1981. and the editor's comment after that note.). Now this result is called ''Gallager- Ryabko Theorem '' in many papers.
1980---
the "bookstack" code was suggested and published (This code was re-discovered by Bently J.L., Sleator D.D.,Terjan R.E., Wei V.K. and often it is called "move-to-front" code, see my letter in : Comm.ACM,v.30,n9,p.792,1987.).
1984-
The '' Twice- universal code'' was suggested. The code can compress sources with unknown probabilities and unknown memory (or connectivity).
1986---
The theory of coding of combinatorial sources was suggested. (a combinatorial source is defined as the set of infinite sequences whereas probability distribution are not considered. For ex., the set of all infinite text in English (or C++) is the combinatorial source). It is shown that the Hausdorff Dimension plays a rule of the Shannon entropy for combinatorial sources (when the entropy does not exist).
1988---
It was shown that the problem of universal coding and (universal) prediction are very close from mathematical point of view. In fact, it was the first paper when universal prediction was considered. Besides, there was proved that it was impossible to predict the output of an ergodic source with precision, which goes to 0 monotonically. (Later it was called as an open problem by T. Cover in his book ).
1989 ---
The new structure for maintaining of commutative probabilities was suggested in my papers [12-14]. Later this structure and algorithm were applied to the arithmetic code by Fenwick, Moffat and others and now the 'standard' arithmetic code is based on that structure. (See Moffat, Neal and Witten, "Arithmetic coding revisited." ACM Trans Information Systems 16(3):pp. 256-294; ).
1990-2002
The new fast and efficient algorithms have been suggested for homophonic codes , low- entropy sources and many other codes and problems. (See the list of publications). The new approach to prediction and source coding, which is based on Hausdorff Dimension and Complexity Theory, was suggested and developed.

COMPLEXITY OF ALGORITHMS
There were discovered fast algorithms for enumeration of combinatorial objects (permutations, combinations, etc.) and fast method for transforming continued fractions into ordinary fractions. The speed of the suggested algorithms is exponentially superior to known ones.(see [29,30] ).
MATHIMATICAL BIOLOGY
The main difficulties in the analysis of animal ''languages'' appear to be methodological. Many workers have tried to directly decipher animal language by looking for 'letters' and 'words' and by compiling 'dictionaries'. The fact that investigators have managed to compile such for a few species only, seems to indicate not that other animals lack 'languages', but that adequate methods are lacking.
Prof. Zh.Reznikova and me suggested the principally new approach to study quantitative characteristics of communicative systems and important properties of animal intelligence. The main point of this approach is not to decipher signals but to investigate just the process of information transmission by measuring time duration which the animals spend on transmitting messages of definite lengths and complexities. This allows to estimate intellectual potentials by observing the communicative process. Our experiments based on ideas of Information Theory have shown that ants probably have an even more intricate form of communication than the honeybee. We also succeeded in studying some properties of insect cognitive capacities , namely their ability to perform limited counting and to memorize simple regularities, thus compressing the information available. The analysis of time duration of ants' contacts enable us to create a hypothesis of how they use numbers and coordinates in their communication. At last, the ants seem to be able to add and subtract small numbers and this result is proved due to using of ideas of Information Theory.