This dissertion makes use of the Legendre symbol, Hilbert norm residue symbol, and the Hasse-Minkowski invariant in proving the nonexistence of some incomplete block designs. A development of nonexistence proofs of certain balanced incomplete block designs is presented which leads to simplifications of some of the necessary conditions for symmetrical balanced incomplete block designs. Extensions of these tables of designs with replications from 11 to 15, similar to those of D.A. Sprott's tables (Sankhya, 1962) on balanced incomplete designs, are provided for symmetrical partially balanced designs with two-associate classes under the group divisible, triangular and latin square schemes. In all, 711 parameter combinations which satisfy the parameter relations are studied, and of the 711 possible designs, 333 have been shown not to exist. Finally, some necessary conditions are worked out for the existence of intra- and inter-group balanced incomplete block designs, an are in which relatively much less is known subsequent to the definition provided by Nair and Bose (Science and Culture, 1942). In this category, 93 possible designs have been studied, of which 72 have been shown not to exist.