In this thesis we present a constructive method for the recovery of an unknown potential q(x) in a new family of inverse Sturm-Liouville problems. One of the members of this family is the following:;Let \[lambda][subscript]n\[subscript]spn=0[infinity] be the eigenvalues of the differential equation -y[superscript]''+q(x)y=[lambda] ysubject to the boundary conditions \eqaligny(0) - hy[superscript]'(0) &= 0 y(1) + Hy[superscript]'(1) &= 0 where the potential q(x) is known to be anti-symmetric about the point x=1[over]2. Can we obtain q from \[lambda][subscript]n\[subscript]spn=0[infinity]?;The main idea is to use the spectral data to synthesise the boundary data for a certain Goursat problem and then to use a time domain technique. We prove a uniqueness theorem to the family of inverse problems. We also prove a theorem about the convergence of the method and give some numerical examples.