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Since Moldova used to be a part of both Romania and Russia, it shares the language and traditions of both countries. In order to suppress any attempts to rejoin Romania and to eliminate most of the links with it after 1940, Moldova was heavily 'russianised'. Many native Moldovans were deported to Siberia for no reason. Even the language was subject to 'russianisation' as well. School textbooks were printed in a language that had not existed before - Romanian words were substituted with new, effectively Russian ones but with Romanian suffixes and prefixes. Cyrillic alphabet was introduced to take over the Latin, that had been used for the last few centuries. The language itself was renamed into 'Moldavian', although it never really changed (it has become more of a dialect over the years). This can account for the fact that the language in Moldova is very muddled now - it's a mixture of Russian and Romanian.
The heavy suppresion of Moldovan/Romanian ethnicity has resulted in a big void in the current self-perception of the Moldovan people as a nation. Few people can unambiguously identify themselves as Romanians. Somewhat more people think they are Moldovans (although in my opinion we are Moldovan Romanians, but not just Moldovans), the language has even lost its name: it is officially called Moldovan, but it is Romanian (just think of the relationship between the Romanian language in Moldova to the one in Romania as between the English language spoken in the USA and in UK). In late 2004 there was a census in Moldova. One of the obvious questions was asking about ethnicity. Foreign observers recorded cases when some of the individuals conducting the census actually told the citizens not to choose 'Romanian' and say 'Moldovan' instead.
In the last year of the Soviet Union's existence, on August 31, 1989, Latin alphabet was reintroduced to replace the artificially imposed Cyrillic and this date is now celebrated as the 'Limba Noastra' ('Our Language') day. But it will take more than a symbolic date to give people their nationality back.
In terms of traditions, most customs are Romanian, but urban culture tends to be more Russian. That's why Russian language has become de facto the universal business language in Moldova, but in villages mainly Romanian is spoken (although most people in villages still can't use the Latin alphabet and use the one they had learnt in school decades ago - the Russian).
In the south of the country Russian language is official, because the Gagauz ethnic minority has its own language - the Gagauz, which has its roots in the Turkish language, spoken by the Ottomans who occupied this area some several hundred years ago.
The folk culture luckily has been preserved and is essentially Romanian (tales, legends, music, dance etc.).