Regarding Rwanda, Burundi might as well be called Rwanda's Lesser Star twin brother, what with them speaking the same language (Rwanda-Rundi) and being one and the same for centuries until sometime during the colonial era, in which their kings voted to separate it into two different realms. The two were ruled by the same European masters (Imperial Germany and, after World War I, Belgium), suffered under the same rather arbitrary classification between their ethnic group (Hutus and Tutsis), retained their monarchies well throughout the colonial period until a few years after independence, and, unfortunately, had the same ethnic tensions plaguing the aforementioned Hutus and Tutsis.
Long story short, the period following independence was characterized by long, long arguments regarding what ethnic group is fit to control the nation. The Tutsis ruled as a dominant minority despite numbering only 10%, which predictably angered the much more-dominant Hutus, who started rebellions in the frontier regions and pogroms directed against the Tutsis, who retaliated with their own pogroms. So just like Rwanda all over again. The reason why Burundi is called the lesser star, however, is the fact that the pogroms never quite reach the same intensity as the genocide that claimed an estimated 1 million people in Rwanda; the pogroms are classified as genocides by a UN report, but it "only" claimed about twenty thousands or so.
Eventually, the UN and the neighboring African countries' peacekeepers (most notably from South Africa) had to step down and help mediate a power-sharing agreement in order to give the Hutsus more representation, which (sort of) worked, as the current president (Pierre Nkurunziza) is a Hutu. Remnants of the Hutu rebel groups still sometime commit terror attacks against the country, though. Recently, the country has come under fire for criminalizing homosexuality.
The Burundian flag