For a (smooth connected) variety $X$ over the complex numbers, the topological space $X(\mathbb C)$ is homotopy equivalent to a finite CW-complex, and thus its fundamental group $\pi_1(X(\mathbb C))$ admits a finite presentation. This implies that the étale fundamental group $\pi^{\rm ét}_1(X))$, which is a profinite group, is topologically finitely presented. In characteristic $p>0$, the picture is much more subtle due to the abundance of Artin-Schreier covers of affine schemes. Grothendieck proved (topological) finite generation of $\pi^{\rm ét}_1(X)$ for proper $X$ over $\bar{\mathbb F}_p$ and Esnault-Shusterman-Srinivas (with a small remark by L.-Srinivas-Stix to relax the assumptions) have recently shown finite presentation. We continue this study in the rigid (i.e. analytic over some non-archimedean field) and log-geometric settings. For this, we introduce the appropriate tame étale fundamental group $\pi^t_1(X/K)$ of a rigid space. The proofs rely on techniques of logarithmic geometry (extended beyond its usual scope of finitely generated monoids). This is joint work with P. Achinger, K. Hübner and J. Stix.