We briefly review the framework of fragmenting jet functions (FJFs) introduced in soft collinear effective theory (SCET) to describe the fragmentation of an identified hadron within a jet. We illustrate how FJFs can be integrated with the leading power (LP) factorization in non-relativistic quantum chromodynamics (NRQCD) in studies of quarkonium production within jets. We present recent theoretical, experimental, and computational developments. We also present the framework of transverse momentum dependent (TMD) FJFs which describe the fragmentation of a hadron within a jet and the transverse momentum of the identified hadron is measured with respect to the jet axis. We show how the measurement of the transverse momentum spectrum can help discriminate between the leading production mechanisms of J/ψ.