Modified from an earlier answer of mine

The Soviets did bring mass literacy, but in a qualified sense of the word. Soviet gains in education were definitely real and the state put great stock in both achieving mass literacy and opening up the upper echelons of education to social groups excluded from education under the tsars. Lenin noted that "the illiterate person stands outside of politics," and placed a high priority upon achieving mass literacy even during the Civil War, and under the leadership of Anatoly Lunacharsky, Narkompros (the People's Commissariat for Education) devoted a large mount of resources to mass literacy and restoring the educational system of the Soviet Union into a semblance of order after the disruption caused by revolution and civil war. The task faced by Narkompros was quite daunting; not only did they have to educate the masses, but also train an entire new cadre of teachers, and vet textbooks and pedagogy to suit the ideas of the new order. By 1924, there state had put in plans for compulsory universal education, as well as coordination between Narkompros and other organs of the Soviet state like the Zhenotdel. The results of mass mobilization around literacy and shock action in education was a marked drop in illiteracy rates, which Lunacharsky would often trumpet. The class character of the Soviet university system also began to change as well, with Narkompros encouraging a type of class-based affirmative action in which proletarian students received preferential treatment and stipends to pay for tuition. The Soviet university system also began to turn out the specialists needed to build socialism as well.

But it is very easy to overstate these successes of the USSR in the field of education. Although Soviet authorities often cast a lot of shade on tsarist literacy rates, noting that in the 1897 census only one in five Russian subjects could sign their name, the fact was literacy rates were markedly climbing in the late tsarist period. Education and other forms of popular literacy were on an clear upward trend that the war disrupted. The common image of a vast number of illiterate peasants was one of the stock tropes of Russian backwardness for the late imperial period, but the actual state of literacy in imperial Russia was more complex. In terms of the other great powers during the fin de siècle, Russian literacy was definitely below par. The 1897 census estimated that approximately 33% of men and 14% of women in European Russia were literate. Yet, despite the continued presence of illiteracy inside Russia, there was a marked uptick in literacy by a number of metrics. The percentage of troops who were literate entering into the Russian Army in 1874 was 21% but had climbed up to 68% in 1913. By contrast, literacy in the 1870s British, French, and German armies hovered around 80%, which demonstrates not only how much Russian literacy had improved, but how far it still needed to go. Despite much hand-wringing among the obshchestvennost' (educated society) about the inability of the peasantry to buckle down and keep to their education, there were strong signs that older habits of prematurely ending a student's education were on the wane. Zemstvo commissions on education and teacher surveys did show that Russian primary school teachers did think their students were eager to learn and become literate. These sentiments are borne out by the mushrooming of popular peasant literature, often in the form of cheap broadsheets and popular literature and Russian publishing transformed from a cottage industry into something more akin to its contemporaries in Western Europe, the US, and Japan. The number of periodical journals and newspapers also grew in the growing Russian urban sphere and the easing of censorship after 1905 allowed for the press to cater to a greater variety of topics. Russian fiction also had a tendency to cater to the masses instead of the obshchestvennost'.

When examining the data produced by Narkompros, much of its gains in literacy came not from Soviet schoolchildren, but from adult cohorts who were the beneficiaries of educational programs during the war and civil war. The fiscal retrenchment of the NEP period, often valorized as the golden age of Lenin, meant Soviet schools were often denuded of funding and resources. A good deal of internal Narkompros memoranda often show a frustration with the inability to squeeze funding from the state as well as frustration that local teachers often ignored directives from the centre. Access to university education was also chaotic and often quite unfair; stipends for proletarian students could not meet the needs of tuition and the early Soviet university system still privileged those from a white-collar technical background who had other means of non-state support.

Further complicating the image of a golden age Lenin and a dark Stalin turn, Soviet education only became more homogeneous and universal under Stalin. In contrast to the parsimony of NEP, the Stalinist turn lavished funding on education, more than doubling the base number of teachers between 1928-38 and expanding higher education. The state placed a renewed effort on teacher certification, in contrast to the more wildcat days of the NEP. Although Stalinist education, like other aspects of the Stalinist system, had a strong coercive element to it, the state's renewed emphasis upon education did have significant mass appeal. Popular worker magazines often had advice columns for how to be kulturnost' (cultured) with questions like "which works of Shakespeare should I know?", showing that some of the interest in education emanated from below.

Yet the Stalinist educational drive did long-lasting damage to the state of Soviet education that the postwar years could only fitfully repair. Stalinist pedagogy stressed that children were adults in miniature, and thus classrooms were largely inflexible environments as teachers sought to maintain discipline and enforce rote learning. Soviet propaganda on education often stressed a more coercive approach to education, such as this famous Soviet adult anti-illiteracy poster which uses guilt (the text reads "mother, if you could read, you would help me") to shame its targets, or various shock/front approaches to education in which short-term herculean efforts were prized over more quotidian methods. Although the Stalin period ushered in mass literacy arguably for the first time in Russian history, the mass public soon found itself with very little they wanted to read. Although the Soviet reading public still had a strong undercurrent of kulturnost' and a respect for the Russian canon, literary tastes shifted to less elevated genres that state-run cooperatives for writers did not stress. The problem of reading shortages became acute in the postwar period where despite efforts by the state to produce the most books in the world, mass tastes in literature did not square with the ideological and politicized literature produced by state presses. Surveys in the 1960s showed that Soviet readers preferred "undemanding" literature, mirroring global trends in literary tastes that stressed reading as a form of escape and pleasure rather than uplift. Thus the postwar Soviet reader was often book-starved amidst plenty of printed materials they did not want to read. Outside of Gorky, the Soviet literary establishment did not produce much of a domestic canon of note.

In a similar vein, university education had long-lasting damage from the Stalin period. The Janus-faced emphasis upon the practical sciences as well as ideologically-charged arts imparted to Soviet higher education a bifurcated character of advanced sciences and liberal arts programs that sought to fit research into Marxist-Leninist paradigms. University education also became a hallmark of privileged status and although nominally open to all, acquiring an advanced education often depended on access to private channels. So-called "boiler-room" jobs became typical for Soviet graduate students by the 1970s as they were jobs that provided both pay and free time, but one could only get such a job through personal connections. The large and clumsy university apparatus the Soviet state inherited from Stalin meant that official forms of support were often inadequate for Soviet students, mirroring the larger problems of Soviet society in which rhetoric often failed to live up to reality.

Sources

Arnove, Robert F., and Harvey J. Graff. National Literacy Campaigns: Historical and Comparative Perspectives. New York: Plenum Press, 1987.

Brooks, Jeffrey. When Russia Learned to Read: Literacy and Popular Literature, 1861-1917. Princeton, N.J.: Princeton University Press, 1985.

Eklof, Ben. Russian Peasant Schools: Officialdom, Village Culture, and Popular Pedagogy, 1861-1914. Berkeley: University of California Press, 1986.