Kardar, Parisi and Zhang introduced a universality class (the 
so-called KPZ universality class) in 1986 which is believed to explain the 
universal behaviour in a large class of two dimensional random growth models 
including first and last passage  percolation. A number of  breakthroughs has 
led to  an explosion of mathematically  rigorous results in this field in 
recent years.  However, these have mostly been restricted to the class of 
"exactly solvable models", where exact formulae are available using powerful 
tools of random matrices,  algebraic combinatorics and representation theory; 
 beyond this class the  understanding remains rather limited. I shall talk 
about a geometric  approach to these problems based on studying the geometry of 
geodesics  (optimal paths), and describe some recent progress along these 
lines.