Abstract: Ballistic capture transfer occurs when a particle becomes captured into orbit about one of the primary mass points of the three or four body problem. Capture in this case is defined when the Kepler two-body energy of particle with respect to the primary is negative. This was first numerically demonstrated by E. Belbruno(EB) in 1986. This capture occurs in a region about the primary called a weak stability boundary(WSB). It was proven by EB (2005) that the WSB contains within it a hyperbolic invariant set. Although crudely numerically estimated in 1986, numerical work by Garcia and G. Gomez(2007) gave deeper insights into its makeup giving surprising results as to its complexity. Further insights in the makeup of this region were made in a series of studies from 2010-2012 by M. Gidea, F. Topputo, EB. The relationship of ballistic capture to permanent capture and the work of K. Sitnikov and J. Moser is discussed. Ballistic capture transfer has important applications. One such transfer was used in 1991 by J. Miller and EB to rescue an errand Japanese lunar probe and successfully bring it to the Moon. This same transfer class was used in 2011 for NASAs lunar Grail mission, and another type was used by ESAs lunar SMART 1 mission in 2004. A new result that is described is the existence of a ballistic capture transfer to Mars that was developed by F. Topputo and EB (2014). This promises to have interesting applications. If there is time, the use of ballistic capture transfers in open star clusters is described which shed light on the Lithopanspermia Hypothesis and the origin of life on Earth(EB, A. Moro-Martin, R. Malhotra, D. Savransky, 2012).