Cell migration is a fundamentally important phenomenon
underlying wound healing, tissue development, immune response
and cancer metastasis. Understanding basic physics of the
cell migration presented a great challenge until, in the last
three decades, a combination of biological, biophysical and
mathematical approaches shed light on basic mechanisms of
the cell migration. I will describe two models, based on
nonlinear partial differential equations and free boundary problems,
which predicted that individual cells do not linger in a symmetric
stationary state for too long, but rather spontaneously break
symmetry and initiate motility. The cells can either crawl straight,
or turn, depending on mechanical parameters. I will show how
experimental data supported the models, and I will also review
current computational challenges.