Communities of Finns in Sweden can be traced back to the Reformation when the Finnish Church in Stockholm was founded in 1533, although earlier migration, and migration to other cities in present-day Sweden, remain undisputed. (Strictly speaking this was not a case of emigration/immigration but of "internal migration" within pre-1808 Sweden.)
In the 16th and the 17th Century large groups of Savonians moved from Finland to Dalecarlia, Bergslagen and other provinces where their slash and burn cultivation was suitable. This was part of an effort of the Swedish king Gustav Vasa, and his successors, to expand agriculture to these uninhabited parts of the country which were later on known as "Finn woods" (Finnskogar).
In Swedish usage the term denominates primarily the indigenous minority of ethnic Finns who ended up on the "wrong" side of the border when Sweden was partitioned in 1809, and the Russian Grand duchy of Finland was created. Cultural imperialism in combination with fear of Russia led to efforts by Sweden's government aiming at assimilation and Swedification of the Finnish speaking population. Similar attempts were initiated already in the late 17th century, but peaked 1850-1950. Finnish speakers remains only along the border to Finland in the furthest North, and as domestic migrants due to unemployment in the North. Depending on definition they are reported to number to 30,000-90,000 - that is up to 1% of Sweden's population. Since the 1970s efforts have been made to reverse some of the effects of Swedification, notably education and public broadcasts in Finnish, to raise the status of Finnish. As a result a written standard of the local dialect Mešnkieli has been established and taught, which has given reason to critical remarks from Finland, along the line that standard Finnish would be of more use for the pupils.
In the 1950s and 1960s the migration from Finland to Sweden was considerable, chiefly due to Finland's misfortune and Sweden's fortune in World War II. The emigration caused some alarm in Finland, particularly when it was reported to approach the level of 10% of the Finns, although a great deal of them returned to Finland in the succeeding decades.
In the Finnish mindset the term "Sweden Finn" (Ruotsinsuomalaiset) is first and foremost directed at these immigrants and their offspring, who at the end of the 20th century numbered at almost 200,000 first-generation immigrants, and about 250,000 second-generation immigrants. Of these 430,000 some 250,000 are estimated to use Finnish in their daily lives, and 100,000 remain citizens of Finland. This usage isn't quite embraced in Sweden.
See also: Sweden-Finnish Language Council