Summary: For a kernel \(k\) in \(L^1(\mathbb{R}^n)\), the spectrum and the essential spectrum of the corresponding convolution operator on \(L^p(\mathbb{R}^n)\), \(1\leq p\leq\infty\), are shown to coincide with the closure of the set \(\{(2\pi)^{n/2}\widehat k(\xi): \xi\in\mathbb{R}^n\}\) in the set of all complex numbers, where \(\widehat k\) is the Fourier transform of \(k\). Under an additional condition on \(k\), a more in-depth study of the spectrum of the corresponding convolution operator on \(L^p(\mathbb{R}^n)\), \(1\leq p\leq\infty\), is carried out. The asymptotic stability of the zero equilibriium solution of a semilinear evolution equation governed by a convolution operator on \(L^p(\mathbb{R}^n)\), \(1\leq p\leq\infty\), is studied.