I will begin by reviewing the history of the stable homotopy theory approach to understanding (co)bordism groups via the Pontrjagin-Thom isomorphism and analysis of the homotopy type of the associated Thom spectra. This proved remarkably successful and led to the determination of many important classical examples of bordism groups such as unoriented, oriented, unitary, special unitary, spin and spin^c. The two outstanding classical cases which are not completely understood are framed bordism (aka stable homotopy groups of spheres) and symplectic bordism. More recently attention has turned to String bordism and its variants.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3444
[3] http://www.mpim-bonn.mpg.de/de/TopologySeminar