  The currently accepted mathematical description of the fundamental
constituents and interactions of matter is the Standard Model of particle
physics. Its last missing particle, the famous Higgs boson, was observed at the
Large Hadron Collider at CERN in 2012. However, it is clear that the Standard
Model cannot be the ultimate theory of Nature, and e.g. cannot account for Dark
Matter or non-vanishing neutrino masses (and does not include gravity). In
fact, searches for physics beyond the SM have been intensified since the Higgs
boson discovery. In this article, we review the hints for new physics, called
``anomalies'', obtained in particle physics experiments within the last years.
We consider both direct high-energy searches for new resonances at the LHC and
indirect low-energy precision experiments. These anomalies range from the
nuclear scale (approximately the mass of the proton) to the electroweak scale
(i.e. the mass of the Higgs boson) to the TeV scale (the highest scale directly
accessible at the LHC), therefore spanning over four orders of magnitude. After
discussing the experimental and theoretical status of the anomalies, we
summarize possible explanations in terms of new particles and new interactions.
In particular, new Higgs bosons and leptoquarks are promising candidates.
Discovery prospects and implications for future colliders are discussed.
