Many real-world applications, although being non-linear, can be well described by
linearized models. Therefore, Linear Programming (LP) became a widely studied and
 applied technique in many areas of science, industry and economy. Semidefinite
Programming (SDP) is an extension of LP. A matrix-variable is optimized over the
intersection of the cone of positive semidefinite matrices with an affine space.
It turned out, that SDP can provide significantly stronger practical results than
 LP. Since then SDP turned out to be practical in a lot of different areas, like
  combinatorial optimization, control theory, engineering, and more recently in
 polynomial optimization. \\ Due to the numerous areas of applications, solving
SDPs became a widely studied subject. Interior-Point Methods are the most popular
algorithms nowadays. Recently, bundle methods and applying an augmented Lagrangian
algorithm have been used for solving SDPs. \\ In this talk I will state the basic
 concept of SDP. I will show how to apply SDP to approximate the max-cut problem
 and the coloring problem, two combinatorial optimization problems. Via the SDP
 relaxations of these two combinatorial optimization problems I will explain the
methods for solving SDPs, in particular the recently developed bundle method for
    max-cut and the boundary point method to compute the Lovasz theta-number.