This talk surveys some recent work in algebraic combinatorics that illustrates the surprising connections between pure algebra and classical combinatorics. The algebraic objects of interest here are the spaces of diagonal harmonics introduced and studied by Garsia, Haiman, et al., which consist of polynomials satisfying certain partial differential equations. The combinatorial objects of interest are lattice paths and parking functions. J. Haglund and the present author have discovered statistics on these objects that are conjectured to give the Hilbert series of diagonal harmonics and related spaces. These statistics have many remarkable combinatorial properties, which have led to new results and open problems in enumerative combinatorics.