Every locally compact group $G$ has a Fourier transform which acts on the convolution algebra of Haar integrable functions on $G$, denoted $L^1(G)$. In the case that $G$ is the group of real numbers or integers, this Fourier transform is the usual Fourier transform studied in undergraduate analysis. In a classical paper of Norbert Wiener from 1930, he shows that a proper ideal $I \subset L^1(\mathbb{R}^d)$ is dense if and only if the Fourier transform of every element of the ideal $I$ vanishes nowhere. It is a big question in Banach algebra theory to determine for which other locally compact groups $G$ does $L^1(G)$ have this property. Groups which do possess this property are called “Wiener” groups, and in some sense, their Fourier transform resembles the Fourier transform on $\mathbb{R}^d$. In this talk I will give an introduction to this topic, and report on recent work, where I show that many totally disconnected locally compact groups are not Wiener.