We introduce vector-valued Jacobi-like forms associated to a representation \(\rho: \Gamma \rightarrow GL(n,{\Bbb C})\) of a discrete subgroup \(\Gamma \subset SL(2,{\Bbb C})\) in \({\Bbb C}^n\) and establish a correspondence between such vector-valued Jacobi-like forms and sequences of vector-valued modular forms of different weights with respect to ρ. We determine a lifting of vector-valued modular forms to vector-valued Jacobi-like forms as well as a lifting of scalar-valued Jacobi-like forms to vector-valued Jacobi-like forms. We also construct Rankin-Cohen brackets for vector-valued modular forms.