Dropping the symmetry axiom from the definition of a distance, we obtain the definition of a quasidistance. Matrices whose entries are quasidistances among a finite set of points are called quasidistance matrices. A digraph with valued arcs realizes a quasidistance matrix when, for a certain subset $\{ 1, \cdots ,n\} $ of its vertices, the shortest path from an arbitrary vertex i to an arbitrary vertex j has a length equal to the entry $d_{ij} $ of the matrix. We give the optimal and suboptimal digraph realizations of quasidistance matrices of order 2 and 3 (except for a special class); these are realizations whose total length is as small as possible. We characterize quasidistance matrices which have optimal realizations. We prove several results relating quasidistance matrices to their principal submatrices and to matrices obtained by decreasing all nondiagonal entries in a column or in a row. We investigate and characterize a class of quasidistance matrices, to be called shrinkable, which by their definition and properties may be considered the analogues of tree-realizable (symmetric) distance matrices.