Loosely-bound objects such as light nuclei are copiously produced in proton-proton and nuclear collisions at the Large Hadron Collider (LHC), despite the fact that typical energy scales in such collisions exceed the binding energy of the objects by orders of magnitude. In this review we summarise the experimental observations, put them into context of previous studies at lower energies, and discuss the underlying physics. Most of the data discussed here were taken by the ALICE Collaboration during LHC Run1, which started in 2009 and ended in 2013. Specifically we focus on the production of (anti-)nuclei and (anti-)hypernuclei. Also included are searches for exotic objects like the H-dibaryon, a possible uuddss hexaquark state, or also a possible bound state of a Λ hyperon and a neutron. Furthermore, the study of hyperon-nucleon and hyperonhyperon interactions through measurements of correlations are briefly discussed, especially in connection with the possible existence of loosely-bound states composed of these baryons. In addition, some results in the strange and charmed hadron sector are presented, to show the capabilities for future measurements on loosely-bound objects in this direction. Finally, perspectives are given for measurements in the currently ongoing Run2 period of the LHC and in the future LHC Run3.