Is there a particular period of Islamic history that you're interested in? Or would you prefer one book that gives an overview from 600AD - the 20th century? In either case, I would probably recommend:

In God's Path by Robert Hoyland. This book is a pretty engaging account of the origins and spread of the Islamic Empire. How did warring tribes from Arabia suddenly conquer territory from Iran to Spain? How did an Islamic identity emerge? And other such interesting questions.

A second, more pop-history style book would be No God But God by Reza Aslan. Aslan is more politically charged than Hoyland, talking about the West's relationship with Islam and its place in the future. Of the three I'm recommending, this one definitely relates the most to current affairs.

For a more Academic-style of book, A History of Islamic Societies by Ira Lapidus is the classic primer for those interested in Islamic history. It is definitely more comprehensive than the other two, stretching most of Islamic history. Depending on your interests you can skip to various sections, but otherwise it would be a bit of a challenge at near 900 pages... If you want just one book though, I definitely recommend this one as the way to go.

Islamic Empires

If you're willing to go beyond single volume, the classic study of Islamic empires is the three volume The Venture of Islam by Marshall Hodgson.

Any book recommendation about History of Scandinavian countries? (Sweden, Denmark, Norway)