Multiterawatt femtosecond duration laser pulses, when launched in the atmosphere, undergo extreme self-focusing to produce dramatic spectral superbroadening and breakdown the molecular constituents of air to produce extended electron-ion plasma channels. The dramatic nonlinear events occurring in the interaction zone while only qualitatively understood, have prompted a flurry of activity over the past decade into exploiting these phenomena in real-life applications. Some applications include femtosecond atmospheric lidar, remote detection of pollutants and/or chem/bio agents, remote laser induced breakdown spectroscopy (LIBS), control of lightning and remote terahertz spectroscopy. The University of Arizona is the lead institution in a new Multi-disciplinary University Research Initiative (MURI) project along with five other institutions. The goal is to gain a deeper understanding of the complex nonlinear physics involved and build mathematically rigorous and computationally feasible models of intense pulse filamentation in air and condensed media. In this talk, I will present a brief history of the development of this emerging field of nonlinear science, point to some successes achieved along the way and outline the strategy of the MURI team aimed at solving remaining outstanding problems. The Arizona team has been to the forefront in developing the present theoretical foundation and more recently we have built a TW femtosecond laser laboratory to support the theory and simulation effort. One intriguing departure from the status quo is to envisage laser beam profiles constituted from linear superpositions of conical waves. Such beams (Bessel and Airy) are remarkably robust under propagation and exhibit a self-healing property which is fundamentally different behavior from that of conventional Gaussian beams.