Due to the accelerated expansion of the visible Universe, the Symmetron has been proposed as a modification of the standard Einstein field equations. I investigate the feasibility of the Symmetron in light of radial perturbations of neutron stars. I derive two second order differential eigenvalue equations describing such perturbations in the Symmetron model, and these equations have not been derived before. These two equations, I solve numerically using the shooting method, and the results show that the presence of the Symmetron makes the star slightly more unstable. This is in contrast to , which claims that the presence of the Symmetron makes the star more stable. I investigate the stability of the star by looking at the numerical value of the fundamental eigenvalue of the two eigenvalue equations. The fundamental eigenvalue differs so slightly from its standard GR limit however, that by solving the eigenvalue equations, I am unsuccessful in constraing the parameters of the Symmetron. Indeed, in light of the investigations undertaken in this thesis, the Symmetron seems completely plausible. Furthermore, from the analytic results, I find a lower limit to one of the key parameters of the Symmetron. This lower limit was, to the best of my knowledge, not known until I discovered it in this thesis.