K-geodetic graphs (K capital) are defined as graphs in which each pair of nonadjacent vertices has at most K paths of minimum length between them. A K-geodetic graph is geodetic if K=1, bigeodetic if K=2 and trigeodetic if K=3. K-geodetic graphs are applied effectively to the solution of several practical problems in distinct areas of computer science, hence the importance of their study. Four problems are central to the study of K-geodetic graphs, namely, characterization, construction, enumeration and classification. The problems of finding the general classification of K-geodetic graphs for each of their classes K=1,2,3 are open. The present paper is a survey dedicated to the construction of K-geodetic graphs for K=1,2,3 using balanced incomplete block designs (BIBDs). To this purpose, we use block designs as combinatorial structures defined in terms of completely predetermined parameters, which is essential for the easy construction of the K-geodetic graphs described in this survey.