Glioblastoma multiforme (GBM) is an aggressive brain cancer that is extremely fatal.  It is characterized by both proliferation and large amounts of migration, which contributes to the didifficulty of treatment.  Based on the so-called go or grow hypothesis, existing models of GBM growth often include two separate equations to model proliferation or migration processes.  Based on a well known in vitro experiment data set of GBM growth, we formulate, validate, simulate, study and compare two plausible models of GBM growth.  We propose first a single equation which uses density dependent diffusion to capture the behavior of both proliferation and migration. We analyze the model to determine the existence of traveling wave solutions.  To prove the viability of the density-dependent diffusion function chosen, we compare our model with the in vitro experimental data. We also present some mathematical open questions. Our second model is build on the Go or Grow hypothesis since glioma cells tend to exhibit a dichotomous behavior: a cell either primarilyproliferates or primarily migrates. We analytically investigate an extreme form of the Go or Grow hypothesis where tumor cell motility and cell proliferation are considered as separate processes. Different solution types are examined via approximate solution of traveling wave equations and we determine conditions for various wave front forms.