Under the standard linear model of the form \[ Y = \boldsymbol{X}^\top \boldsymbol{\theta}^* + \varepsilon, \] the authors are concerned with robust statistical inference methods for the parameter vector \(\boldsymbol{\theta}^*\), provided that a random sample \((Y_1, \boldsymbol{X}_1), \ldots, (Y_n, \boldsymbol{X}_n)\) is at hand. In this, they define robustness as robustness against heavy tails of the regression error term \(\varepsilon\), meaning that only a few finite moments of \(\varepsilon\) exist (conditionally to \(\boldsymbol{X}\)). The authors propose to estimate \(\boldsymbol{\theta}^*\) by means of the Huber estimator \(\widehat{\boldsymbol{\theta}}_\tau\), where the tuning parameter \(\tau\) is called the robustification parameter. Based on \(\widehat{\boldsymbol{\theta}}_\tau\), confidence sets can be constructed by means of the multiplier bootstrap method. The authors establish the validity of the resulting bootstrap scheme under certain conditions regarding the choice of \(\tau\), which should be adapted to the sample size \(n\), the dimension \(d\) of \(\boldsymbol{\theta}^*\), and the number \(\delta \geq 0\), when it is assumed that \(\mathbb{E}\left(|\varepsilon|^{2 + \delta} | \boldsymbol{X}\right)\) is finite. Data-driven procedures for choosing \(\tau\) are also discussed. Finally, the authors consider the case that \(m \gg 1\) regression models are simultaneously under consideration, and that their intercepts shall simultaneously be tested for being zero. The latter problem has applications in empirical finance. A multiplier bootstrap-based variant of the linear step-up test by Y. Benjamini and Y. Hochberg [J. R. Stat. Soc., Ser. B 57, No. 1, 289–300 (1995; Zbl 0809.62014)] for control of the false discovery rate is proposed in this context. The authors’ theoretical results are illustrated by numerical studies based on computer simulations.