In this thesis, we are concerned with some combinatorial designs ---1- rotational (v, 4, 1) designs, referee squares, triple whist tournaments and Room squares. Many of our constructions are Z-cyclic. Chapter 2 is devoted to 1-rotational (v, 4, 1). In Chapter 3, a method of constructing Z-cyclic referee squares is given, and we list Z-cyclic referee squares of side 2n - 1 when n < 17 except 4 and 5. Chapter 4 is devoted to triple whist tournaments. Finally, we construct Z-cyclic Room squares in Chapter 5.