  Description of parton interaction with the hot and dense nuclear matter
created in heavy-ion collisions at high energies is a complex task, which
requires a detailed knowledge of jet production in p+p and d+Au collisions.
Measurements in these collision systems are therefore essential to disentangle
initial state nuclear effects from cold nuclear matter effects, medium-induced
$k_T$ broadening and jet quenching. To obtain complete description of the p+p
(d+Au) collision it is also important to study particle production in the
underlying event. The measured properties of underlying event can be used to
tune the QCD based Monte-Carlo models.
  In this paper some of the recent results on jet and underlying event
properties in p+p and d+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV measured by
the STAR experiment are presented. In particular, the preliminary results on
charged and strange particle fragmentation functions in p+p collisions are
discussed and confronted with PYTHIA simulations. Next, the measurement of
inclusive jet spectrum and di-jet correlations in d+Au collisions is presented
and compared with the measurements in p+p collisions to estimate the size of
cold nuclear matter effects. Finally, the study of underlying event properties
in p+p collisions is shown and compared to PYTHIA simulation.
