In order to trace out with more understanding the consequences of general relativity it is advantageous to have exact solutions of Einstein's field equations which show more detail than the familiar solutions with their high symmetry. In the present investigation, based on the method of separation of variables, all solutions of the field equations for empty space have been found which (1) have the "linked pair" form ${g}_{\mathrm{ij}}=\ifmmode\pm\else\textpm\fi{}{\ensuremath{\delta}}_{\mathrm{ij}}{{A}_{i}}^{2}({x}^{0}, {x}^{1}){{B}_{i}}^{2}({x}^{0}, {x}^{3})$, and which (2) are nondegenerate---so far as could be determined---in the sense that all the ${g}_{\mathrm{ij}}$ cannot be reduced to functions of only two variables. Other solutions have been obtained from the solutions of the above form by interchange of variables. Explicit expressions are given for all twenty nondegenerate solutions, all apparently new. Of degenerate solutions, ten are presented, not all of them new. All thirty solutions are examined with respect to possible physical and geometrical interpretations.