Suppose White has only a rook, bishop, and king, and Black has only a king. This ending is always won. First, we consider the case where the kings are separated. The technique is to force the king to an edge of the board and then to a corner opposite in color to the squares to which the bishop is confined. Consider:
8/8/3k4/4R3/4B3/2K5/8/8 w
White to Move and Win
1. Kb3 (wasting a tempo) Kc7 2. Re6 Kd7 3. Bf3 Kc7 4. Rd6 Kb8 5. Rd7 Kc8 6. Rb7 Kd8 7. Rc7 Ke8 8. Rd7 Kf8 9. Re7 Kg8 10. Rf7 Kh8 11. Bd5 Kg8 12. Be6 Kh8 13. Rh7#
If the kings are connected, a setup is needed to pry the kings apart. The setup must be in a corner colored opposite to the squares the bishop is on. For example, suppose the bishop is on white squares. Then put the king on a1 (or h8) and the rook on b2. Then if the Black king moves to a2, move the bishop to b1 and if the Black king moves to b1, then move the bishop to a2. This will force the Black king out. Consider:
8/8/8/8/8/1kR5/2K1B3/8 w
White to Move and Win
1. Kb1+ Kb2 2. Ka1 Kb1 3. Rc2 Kb2 4. Bd3 Ka2 5. Rb2 Kb1 (or 5. ... Ka3 6. Bc2 Ka2 7. Bb1 etc. as in the mainline) 6. Bc2 Ka2 (or 6. ... Kc1 7. Bb1 etc.) 7. Bb1 Ka3 8. Ba2
Now the kings are separated. The nearest correct corner to mate the Black king in is a1. So, White will evacuate his king and force Black back toward a1. We continue:
8. ... Ka4 9. Kb1 Ka5 10. Rb6 Ka4 11. Kc1 Ka3 12. Kd1 Ka4 13. Rb5 Ka3 14. Bc4 Ka4 15. Bd3 Ka3 16. Rb4 Ka2 17. Rb3 Ka1 18. Bc4 Ka2 19. Ke1 Ka1 20. Rb1#