  In 1990, Lakshmibai and Sandhya published a characterization of singular
Schubert varieties in flag manifolds using the notion of pattern avoidance.
This was the first time pattern avoidance was used to characterize geometrical
properties of Schubert varieties. Their results are very closely related to
work of Haiman, Ryan and Wolper, but Lakshmibai-Sandhya were the first to use
that language exactly. Pattern avoidance in permutations was used historically
by Knuth, Pratt, Tarjan, and others in the 1960's and 1970's to characterize
sorting algorithms in computer science. Lascoux and Sch$\text{\"u}$tzenberger
also used pattern avoidance to characterize vexillary permutations in the
1980's. Now, there are many geometrical properties of Schubert varieties that
use pattern avoidance as a method for characterization including Gorenstein,
factorial, local complete intersections, and properties of Kazhdan-Lusztig
polynomials. These are what we call consequences of the Lakshmibai-Sandhya
theorem. We survey the many beautiful results, generalizations, and remaining
open problems in this area. We highlight the advantages of using pattern
avoidance characterizations in terms of linear time algorithms and the ease of
access to the literature via Tenner's Database of Permutation Pattern
Avoidance. This survey is based on lectures by the second author at Osaka,
Japan 2012 for the Summer School of the Mathematical Society of Japan based on
the topic of Schubert calculus.
