  This thesis consists of two parts. In the first part we study some topics in
$\mathcal{N}=1$ supersymmeric gauge theory and the relation to matrix models.
We review the relevant non-perturbative techniques for computing effective
superpotential, such as Seiberg-Witten curve. Then we review the proposal of
Dijkgraaf and Vafa that relates the glueball superpotentials to the computation
in matrix models. We then consider a case of multi-trace superpotential. We
perform the perturbative computation of glueball superpotential in this case
and explain the subtlety in identifying the glueball superfield. We also use
these techniques to study phases of $\mathcal{N}=1$ gauge theory with flavors.
In the second part we study topics in AdS/CFT correspondence and its plane wave
limit. We review the plane wave geometry and BMN operators that corresponding
to string modes. Then we study string interactions in the case of a highly
curved plane wave background, and demonstrate the agreements between
calculations of string interaction amplitudes in the two dual theories. Finally
we study D3-brane giant gravitons and open string attached to them. Giant
gravitons are non-perturbative objects that have very large R-charge.
