The ancient inhabitants of Bosnia and Herzegovina were Illyrians who overran the
 northwestern part of the Balkan Peninsula. Most of the country was included in the  Roman province of Dalmatia. Slav settlement began in the latter half of the 6th century.  For the next five centuries various parts of the region fell under the rule of Serbs,  Croats, Hungarians, Venetians, and the Byzantines. In the 12th century the Hungarian  kingdom began to rule the region through a series of "bans," or governors, though  Bosnia enjoyed periods of power and independence. In 1322 a Bosnian, Stjepan  Kotromanic, was elected ban, and for the rest of the 14th century the Kotromanic
 dynasty (especially under Stjepan's nephew, Tvrtko I) expanded Bosnia's territory to  include the Serb province of Hum and much of the Adriatic coast.

 The Ottoman Turks invaded Bosnia in 1386, and after many battles it became a Turkish  province in 1463. Hum held out longer under rulers who styled themselves herceg  ("duke") of St. Sava--a name recalled today in Herzegovina. During the 16th and 17th  centuries, Bosnia was an important Turkish outpost in the constant warfare against  the Habsburgs and Venice. During this period much of the native population converted  to Islam. As the Ottoman Empire was pushed out of Europe, its rule in Bosnia became  more onerous, and Muslims and Christians alike grew resentful of interference from  Constantinople.

 At the Congress of Berlin after the Russo-Turkish War of 1877-78, Bosnia and  Herzegovina was assigned to Austro-Hungarian occupation, though it was still  nominally Turkish. It was annexed to Austria-Hungary in October 1908. A new  constitution divided the electorate into three electoral colleges and assigned in each a  fixed proportion of seats to the Orthodox, Roman Catholics, and Muslims. This did little  to satisfy growing Serb nationalism, and on June 28, 1914, the Austrian archduke  Francis Ferdinand was assassinated at Sarajevo by a Bosnian Serb student, Gavrilo
 Princip. This event precipitated World War I. Bosnia and Herzegovina was annexed to  Serbia on Oct. 26, 1918, as part of the Kingdom of Serbs, Croats, and Slovenes. During World War II Bosnian Serbs suffered greatly under the genocidal policies of the Croatian puppet state. In 1946 the twin territory became one of the republics of communist Yugoslavia.

 With the collapse of communism in 1989-90, Bosnia and Herzegovina was engulfed by  a wave of nationalism that swept Yugoslavia. After Croatia quit the federation in 1991,  Bosnian Croats and Bosniacs approved referenda calling for an independent,  multinational republic. The Bosnian Serbs, however, refused to secede from Yugoslavia,  which by now was dominated by Serbia, and from 1992 Bosnia and Herzegovina was  wracked by a war in which entire populations were "cleansed" from areas taken over  by each nationality. Serbs seized much of the north and east, Croats took the west,  and Bosniacs held onto cities in the centre and northwest. The war ended in December  1995 with a peace accord that created a loosely federalized Bosnia and Herzegovina  divided roughly evenly between a Bosniac-Croat federation and a Serb republic.

Taken from Britanica.com

                                      The area was part of the Roman province of Illyricum. Bosnia
                                      was settled by Serbs in the 7th century; it appeared as an
                                      independent country by the 12th century but later at times
                                      acknowledged the kings of Hungary as suzerains. Medieval
                                      Bosnia reached the height of its power in the second half of the
                                      14th century, when it controlled many surrounding territories.
                                      Bosnia also annexed the duchy of Hum, which, however,
                                      regained autonomy in 1448 and became known as Hercegovina.
                                      During this period the region was weakened by religious strife
                                      among Roman Catholics, Orthodox, and Bogomils. Thus
                                      disunited, Bosnia fell to the Turks in 1463. Hercegovina held out
                                      until 1482, when it too was occupied and joined administratively
                                      to Bosnia. The nobility and a large part of the peasantry
                                      accepted Islam.

                                      Under Turkish rule, Bosnia and Hercegovina's economy
                                      declined. Physical remoteness facilitated the retention of
                                      medieval social structure, including serfdom (remnants of which
                                      lasted until the 20th century). Refusal by the Turkish to institute
                                      reforms led to a peasant uprising (1875) that soon came to
                                      involve outside powers and led to the Russo-Turkish War of
                                      1877–78. After the war, the Congress of Berlin (1878) placed
                                      Bosnia and Hercegovina under Austro-Hungarian administration
                                      and occupation, while recognizing the sovereignty of the Turkish
                                      sultan. Austria-Hungary improved economic conditions in the
                                      area but sought unsuccessfully to combat rising Serbian
                                      nationalism, which mounted further when Bosnia and
                                      Hercegovina were completely annexed in 1908.

                                      The assassination (1914), by a Serbian nationalist, of Archduke
                                      Francis Ferdinand in Sarajevo precipitated World War I. In
                                      1918 Bosnia and Hercegovina were annexed to Serbia. The
                                      dismemberment of Yugoslavia during World War II led to
                                      Bosnia and Hercegovina's incorporation into the German puppet
                                      state of Croatia. Much partisan guerrilla warfare raged in the
                                      mountains of Bosnia during the war. In 1946, Bosnia and
                                      Hercegovina became one of the six constituent republics of
                                      Yugoslavia.

                                      Following the secession of Slovenia, Croatia, and Macedonia in
                                      1991, the Croats and Muslims of Bosnia and Hercegovina,
                                      fearing Serbian domination, voted for a declaration of
                                      independence from Yugoslavia in Oct., 1991. In Apr., 1992, the
                                      sovereignty of Bosnia and Hercegovina was recognized by the
                                      European Community and the United States. Serb troops, both
                                      from Serbia and the local population, began to carve out the
                                      Serb-populated areas and declared the imprecisely defined
                                      Serbian Republic of Bosnia and Hercegovina. Croats followed
                                      suit and declared the creation of their own Croatian Community
                                      of Herceg-Bosna.

                                      The poorly armed Muslims were forced from their homes and
                                      towns as part of an “ethnic cleansing” policy carried out mostly
                                      by the Serbs. Thousands of Muslims were killed, many were
                                      placed in detention camps, and many more fled the country,
                                      especially to Croatia or to Germany. Only remnants of the
                                      Muslim-dominated government of Bosnia and Hercegovina
                                      remained, and it had almost no authority outside of Sarajevo. In
                                      1992 and 1993 the integrity of the borders of Bosnia and
                                      Hercegovina was threatened by both Croatia and Yugoslavia.

                                      In 1993 Bosnian government forces also fought Muslim rebels
                                      for control of Bihac, a largely Muslim enclave in northwest
                                      Bosnia whose leaders declared the region an autonomous
                                      province and signed peace accords with Serb and Croat forces.
                                      Shelling, mainly by Serb forces, destroyed much of Sarajevo and
                                      laid waste to other cities throughout the countryside.

Taken from Infoplease.com