Boundary critical behaviours of systems at bulk criticality have been studied by condensed matter physicists for decades, in the beginning mostly theoretically, then also experimentally. These studies have revealed that the field of boundary critical phenomena is quite rich. For any universality class of bulk critical behaviour usually several distinct universality classes of boundary critical behaviour exist. To elucidate this, we introduce appropriate lattice models and map them onto boundary field theories describing their scale and conformal invariance on large scales. We show that this mapping can be subtle in some cases. We discuss the role and difference of boundary conditions of the field theories on mesoscopic and large scales, to what extent they are characteristic of --- and hence may be used to identify --- the distinct boundary universality classes. Finally, we explain the characteristic near-boundary asymptotic behaviours they exhibit and mention some open problems.