  Linear arrays of trapped and laser cooled atomic ions are a versatile
platform for studying emergent phenomena in strongly-interacting many-body
systems. Effective spins are encoded in long-lived electronic levels of each
ion and made to interact through laser mediated optical dipole forces. The
advantages of experiments with cold trapped ions, including high spatiotemporal
resolution, decoupling from the external environment, and control over the
system Hamiltonian, are used to measure quantum effects not always accessible
in natural condensed matter samples. In this review we highlight recent work
using trapped ions to explore a variety of non-ergodic phenomena in long-range
interacting spin-models which are heralded by memory of out-of-equilibrium
initial conditions. We observe long-lived memory in static magnetizations for
quenched many-body localization and prethermalization, while memory is
preserved in the periodic oscillations of a driven discrete time crystal state.
