This thesis has featured results in a various topics on Combinatorics. First there are results for various intersection conditions on families of set, and families of permutations and generalisations thereof. There are also results on matching sequenceability for graphs and hypergraphs both general results and results for specific graphs and hypergraphs. This includes work on a newly introduced generalisation r-matching sequenceability. The thesis continues with results on the existence of balanced diagonals in frequency squares, when there are 2 or 3 symbols. An analogue of Wei's duality theorem for (q,m)-demimatroids is also featured. Finally there is a discussion of future work related to the contents of this thesis.