Our model of reality is a random single file line of n people waiting to board an airplane. Each person takes a unit time to sit down, and no one can reach his/her seat until those ahead with earlier seats (e.g., first class) sit. What is the waiting time to board all passengers?
Schensted tableau insertion calculates the answer. This is a method of combinatorics and computational mathematics.
In geometry and representation theory, generalized Schensted algorithms appear in the study of: Kazhdan-Lusztig cells, parameterizations of the irreducible components of the Steinberg variety of triples (and Springer's generalization of this to any real linear reductive group), determinantal ideals (De Concini, Eisenbud and Procesi), and crystal bases (e.g., Leclerc and Thibon).
I'll survey such manifestations, with the hope of indicating what kinds of problems are likely to be relevant to insertion algorithms.
Time permitting, I'll discuss my own experience: a joint project with Buch, Kresch, Shimozono and Tamvakis, where we developed a tableau insertion algorithm to answer questions arising in the study of degeneracy loci of morphisms of vector bundles.