  In this work we introduce {\it boundary time-crystals}. Here {\it continuous}
time-translation symmetry breaking occurs only in a macroscopic fraction of a
many-body quantum system. After introducing their definition and properties, we
analyse in detail a solvable model where an accurate scaling analysis can be
performed. The existence of the boundary time crystals is intimately connected
to the emergence of a time-periodic steady state in the thermodynamic limit of
a many-body open quantum system. We also discuss connections to quantum
synchronisation.
