  In recent years, the systems comprising of bosonic atoms confined to optical
lattices at ultra-cold temperatures have demonstrated tremendous potential to
unveil novel quantum mechanical effects appearing in lattice boson models with
various kinds of interactions. In this progress report, we aim to provide an
exposition to recent advancements in quantum simulations of such systems,
modeled by different "non-standard" Bose-Hubbard models, focusing primarily on
long-range systems with dipole-dipole or cavity-mediated interactions. Through
a carefully curated selection of topics, which includes the emergence of
quantum criticality beyond Landau paradigm, bond-order wave insulators, the
role of interaction-induced tunneling, the influence of transverse confinement
on observed phases, or the effect of cavity-mediated all-to-all interactions,
we report both theoretical and experimental developments from the last few
years. Additionally, we discuss the real-time evolution of systems with
long-range interactions, where sufficiently strong interactions render the
dynamics non-ergodic. And finally to cap our discussions off, we survey recent
experimental achievements in this rapidly evolving field, underscoring its
interdisciplinary significance and potential for groundbreaking discoveries.
