First passage percolation (FPP) is a model of random metric, defined typically on an infinite graph, which randomly distorts the graph distance by assigning independent and identically distributed non-negative lengths to each of the edges. First passage percolation was first introduced on the d dimensional Euclidean lattice with nearest neighbour edges in 1965 by Hammersley and Welsh as a model of fluid flow through inhomogeneous media and has been extensively studied since. In two dimensions, precise physics predictions about the asymptotic behaviour of FPP are available from the theory of Kardar-Parisi-Zhang (KPZ) universality class, but understanding this universality rigorously remains a major challenge. 

The first part of this set of lectures will be a review of some of the classical results of FPP and related literature.  I shall discuss the shape theorem, fluctuation bounds and also discuss some of the conjectures regarding the large scale behaviour of the metric and the geodesics. I shall also discuss how some of these conjectures can be approached under some additional unproven hypotheses which nevertheless are widely expected to be true under mild conditions. Exactly solvable models for which the KPZ predictions can be rigorously verified will also be briefly mentioned. 

I shall also discuss the role of the underlying geometry on various arguments and what knows or expects if the underlying geometry is changed (for example to the Cayley graph of a Gromov hyperbolic group). 

The later part of the lectures will discuss some recent results on certain rotationally invariant models of first passage percolation on the plane.