Abstract: Modular forms are complex-valued functions on the upper-half plane $H$ that have certain properties related to the action of $SL(2,Z)$ on $H$ (or another discrete subgroup of $G=SL(2,R))$. Theory of modular forms has always included modular forms with values in a complex vector space, too. Dimension of spaces of vector-valued automorphic forms has been computed, in certain arithmetic situations, not only for $G=SL(2,R)$ but also for other linear groups (e.g. $G=SU(n,1)$, $Sp(n,1))$. This will be a survey-style talk, with the intention to to have a glimpse at a certain part of the theory of vector-valued automorphic forms, including some classical results and some very recent results.