In this PhD thesis we study the isomorphism problem of bi-Cayley graphs and the related question of classifying finite BCI-groups. More precisely, the following questions/problems are considered:

(i) Find effective, sufficient and necessary conditions for the isomorphism of two cyclic bi-Cayley graphs.
(ii) Which groups are 3-BCI-groups?
(iii) Which cubic bi-Cayley graphs are BCI-graphs?
(iv) Which cyclic balanced configurations have the CI-property?
(v) Analytical enumeration of balanced cyclic configurations.

Problem (i) is solved for tetravalent graphs. Problem (ii) is solved for nilpotent groups. We contribute to Problem (iii) by proving that all connected cubic arc-transitive bi-Cayley graphs over abelian groups are BCI-graphs. Regarding Problem (iv), we prove that all cyclic balanced configurations have the CI-property for which the number of points is either a product of two distinct primes, or a prime power. Regarding Problem (v), we derive a close formula for the number of connected cyclic configurations of type (v_3).