Axion-like particles (ALPs), which are gauge-singlets under the Standard Model (SM), appear in many well-motivated extensions of the SM. Describing the interactions of ALPs with SM fields by means of an effective Lagrangian, we discuss ALP decays into SM particles at one-loop order, including for the first time a calculation of the $atopipipi$ decay rates for ALP masses below a few GeV. We argue that, if the ALP couples to at least some SM particles with couplings of order $(0.01-1) mbox{TeV}^{-1}$, its mass must be above 1 MeV. Taking into account the possibility of a macroscopic ALP decay length, we show that large regions of so far unconstrained parameter space can be explored by searches for the exotic, on-shell Higgs and $Z$ decays $hto Za$, $hto aa$ and $Ztogamma a$ in Run-2 of the LHC with an integrated luminosity of 300 fb$^{-1}$. This includes the parameter space in which ALPs can explain the anomalous magnetic moment of the muon. Considering subsequent ALP decays into photons and charged leptons, we show that the LHC provides unprecedented sensitivity to the ALP-photon and ALP-lepton couplings in the mass region above a few MeV, even if the relevant ALP couplings are loop suppressed and the $atogammagamma$ and $atoell^+ell^-$ branching ratios are significantly less than 1. We also discuss constraints on the ALP parameter space from electroweak precision tests.
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