The Stockwell transform of a function \(f\) on \(\mathbb R=(-\infty,\infty)\) is a hybrid of the Gabor transform and the wavelet transform: \[ S_\varphi f (b,\xi)=(2\pi)^{-1/2} |\xi| \int_{-\infty}^{\infty}e^{-ix\xi} f(x) \overline{\varphi (\xi (x-b))}\, dx, \] \(b \in \mathbb R\), \(\xi \in \mathbb R\). The authors study the boundedness and compactness of localization operators associated to \(S_\varphi\) on \(L^p(\mathbb R)\). The results can be extended to the modified Stockwell transform \( S^s_\varphi f (b,\xi)= |\xi|^{1/s-1}S_\varphi f (b,\xi)\).