Jerry Everard's Introduction to Russian Formalism
This summary draws heavily on the work of Prof L.M. O'Toole and Raman Sheldon.

Russian Formalists considered literature to be a special use of language. As such it was amenable to analysis in and of itself. Peter Steiner considers Russian Formalism to fall into three periods:

the machanical view of language;
the organic view - literature as organism of inter-related parts; and
the the systemic view - literature as a system, or organising principle.
Formalism began near the turn of the Century, emerging in the OPOYAZ group (Society for Poetic Language) as a break with the late romantic tradition of symbolism in literature and Futurism and a number of related movements in the visual arts.

The movement sought a non-prescriptive criticism that was part of a more general move towards making literature more accessable to the masses. Victor Shklovsky introduced the idea of 'making strange' in order to derail passive and uncritical reception of texts.

Shklovsky considered the work of art to be the sum of the formal devices of which it is comprised, thus abolishing the firm distinction between form and content. Later moves to orient criticism towards structure as opposed to form avoided the suggestion of form being something exterior to content.

Under this rubric, form becomes merely the organisation of pre-aesthetic materials. Thus Shklovsky differentiated between fabula (the fable) and syuzhet (plot) in terms of the structuring of what is said. Yurii Tynyanov emphasised the binary methodology favoured by the earlier formalists. Words, for Tynyanov were not essentially 'poetic' or 'prosaic' but rather were coloured by the formal textual context in which they were positioned.

Shklovsky, Tynyanov, Eikhenbaum and Tomashevsky considered the textual work in holistic terms as a complex unity of component parts. The parts were analysed in relation to each other. Those that stood out from the others were considered foregrounded. By establishing a 'scientific' critical practice, with the articulation of structural 'laws' then specific fields of literature could be related to other fields.

In 1928 Tynyanov, with Roman Jakobson published the Theses on Language. These formed the basis for the development of structuralism. These were:
 

Literary science had to have a firm theoretical basis and an accurate terminology.
The structural laws of a specific field of literature had to be established before it was related to other fields.
The evolution of literature must be studied as a system. All evidence, whether literary or non-literary must be analysed functionally.
The distinction between synchrony and diachrony was useful for the study of literature as for language, uncovering systems at each separate stage of development. But the history of systems is also a system; each synchronic system has its own past and future as part of its structure. Therefore the distinction should not be preserved beyond its usefulness.
A synchronic system is not a mere agglomerate of contemporaneous phenomena catalogued. 'Systems' means hierarchical organisation.
The distinction between langue and parole, taken from linguistics, deserves to be developed for literature in order to reveal the principles underlying the relationship between the individual utterance and a prevailing complex of norms.
The analysis of the structural laws of literature should lead to the setting up of a limited number of structural types and evolutionary laws governing those types.
The discovery of the 'immananet laws' of a genre allows one to describe an evolutionary step, but not to explain why this step has been taken by literature and not another. Here the literary must be related to the relevant non-literary facts to find further laws, a 'system of systems'. But still the immanent laws of the individual work had to be enunciated first.
Vladimir Propp was influenced by the Formalists, and his work The Morphology of the Russian Folk Tale provided one of the defining studies of genre, and laid the foundations for French Structuralism, influencing particularly the work of Roland Barthes.

Another contemporary figure, Mikhail Bakhtin, was also influenced by if not directly linked with the Russian Formalists. His contributions to the notion of dialogism and the notion of voice in literary discourse emerged contemporaneously with considerations of sound and rhythmic elements in Formalist analyses. Russian Formalism contributed a number of things to literary theory, including:

placing the study of the actual work at the centre of literary scholarship, rather than looking for authorial biographical links or sociolgical influences, which they considered as peripheral to the text.
They problematised the idea of 'literariness', and usefully addressed the 'form' versus 'content' issue.
They viewed literary history and the eveolution of literary genres as as an internal dynamic process.
They contributed a wealth of analytical techniques to stylistic analysis, including sound patterns, metres and verse forms.
They provided analytical techniques for characterising a range of discursive styles and different modes of story-telling.
Structural Formalism continued for some time into the 1930s in the Prague Linguistic Circle. Some of this group, including Roman Jakobson migrated to the US with the emergence of Nazism. This group went on to influence the development of New Criticism in the 1940s and 1950s.
In other directions, the Bakhtin School combined elements of Formalism with Marxism. It was formalist insofar as it was concerned with the linguistic structure of literary texts, but was marxist in its comitment to the view that language could not be separated from ideology. At the same time it resisted the purely marxist turn insofar as it resisted the view that langauge arose as a reflex of a material socio-economic substructure.