Wide range of models that describe behavior of the early universe can give rise to topological defect networks due to Kibble mechanism. 

Evolution of these defects should give observable astrophysical outcomes in the form of gravitational waves, contribution to the cosmic microwave background and gravitational lensing. Having an accurate model of defects evolution it is possible to make predictions for their observational signals and thereby to open the window to the high energy physics of the early universe when  they  were created. Simulations and semi-analytical concept are main approaches to obtain correct description of topological defect network evolution. Two of these methods together with the question of possible defects origination and current observational restrictions will be covered by this talk.

Additionally, there will be presented a result on the domain wall (2 dimensional defect) network simulations, where we show for the first time that a sufficiently large Hubble damping leads to a linear scaling regime where the network is nonrelativistic and which can be also used as a calibration for semi-analytical approach.