This thesis primarily examines several problems in enumerative combinatorics, particu- larly those related to sets of (binomial) paths on an integer lattice. We present three main results of particular interest: (i) an involution for enumerating osculating lattice paths; (ii) a combinatorial proof of a lemma which enables a purely combinatorial method for enumerating non-intersecting paths; and (iii) a combinatorial proof of another recur- rence which, in addition to the previous lemma, enables combinatorial interpretations of product forms. The result in (i) is the first result for osculating paths which allows an arbitrary number of paths. Previous methods have been found for two or three paths, but none previously known were able to be generalised to the full problem. The method presented here uses an involution on a novel object known as stacks, which are readily enumerable. The fixed points of the involution readily biject to non-trivial sets of osculating lattice paths. A more practical constant term formula is also given, in addition to a summation formula. These results are then applied to various classes of alternating sign matrices. The results in (ii) and (iii) are the first known wholly combinatorial proofs for so-called ‘product forms’. These represent the first step in a combinatorial approach to solving the remainder of this very large class of problems. The work is inspired by a pictorial interpretation of a modified jeu de taquin, and this is also given. Both main proofs are bijections between sets of lattice paths on a triangular geometry, with various markings. Finally, we discuss some more practical uses of combinatorial methods, and apply them to several types of problems in genetics. Two key questions are discussed: the plausibility of theorised structures, and the minimal complexity to satisfy the current data. Only general results are given, although the method admits application to many examples, given sufficient data.