Bosnia is the one place in Europe where Islam, Orthodoxy and Catholicism have coexisted as comparable communities for centuries in the same towns, speaking the same language. The three national names in use, Bosniak, Serb and Croat, are religious categories doing the work of national ones, applied to people whose ancestors made different confessional choices under different empires.
Ottoman rule from 1463 brought Islam and produced a distinctive urban civilisation: the mosque complexes with their endowed schools, libraries, kitchens and bridges, funded by the vakuf endowment system; the tekkes of the Naqshbandi and Mevlevi orders; and a legal and scholarly culture conducted in Ottoman Turkish, Arabic and Persian alongside Bosnian. Conversion was gradual and voluntary over generations rather than forced, and the reasons for it are still argued over, including a long standing but now largely discounted theory tying it to the medieval Bosnian Church.
That medieval church is a genuine puzzle. It existed, it was independent of both Rome and Constantinople, it was condemned as heretical by Rome, and what it actually taught cannot be established from the surviving evidence. The dualist Bogomil identification popularised in the twentieth century is not supported by the primary sources, and the honest position is that the doctrine of the Crkva bosanska is unresolved.
The Islamic Community of Bosnia and Herzegovina, headed by a Reis ul ulema, has been an organised, state recognised body since Austro Hungarian rule, and it remains the model for institutional European Islam that other countries study.