We present in this review a consistent picture of the current status o
f epitaxial growth physics, as related to molecular beam epitaxy. Afte
r a short introduction in the field we describe the different growth m
odes and the attributed theoretical models. Among the simulation metho
ds used presently to describe the different growth modes, we selected
the Monte-Carlo simulation for a more detailed example. The experiment
al part of the paper is divided in to two main sections of lattice-mat
ched and lattice-mismatched systems. In the first part, we concentrate
on the growth mechanisms on nominally oriented substrates and substra
tes with vicinal surfaces, which includes the description of the growt
h of tilted and serpentine superlattices. In the second part we discus
s the concept of the critical layer thickness together with island for
mation and strain-induced effects. Besides these two main topics we re
port on the application of surface-active species, which provide a new
avenue to achieve high-quality man-made microstructures against therm
odynamic odds. Within this context self-organisation of epitaxial depo
sits is discussed. Finally, we describe material-related growth peculi
arities, which are usually connected with a specific property of the c
onstituent element, like volatility or extraordinary chemical reactivi
ty. We briefly present the currently known growth characteristics for
IV-IV heterostructures, wide-gap III-V and narrow-gap II-VI compounds.