This dissertation is a study of the relationship between a topological space X and varioushigher-order objects that we can associate with X. In particular the focus is on C(X), the setof all continuous real-valued functions on X endowed with the topology of pointwise convergence,the compact-open topology and an admissible topology. The topological propertiesof continuous function universals and zero set universals are also examined. The topologicalproperties studied can be divided into three types (i) compactness type properties, (ii) chainconditions and (iii) sequential type properties.The dissertation begins with some general results on universals describing methods ofconstructing universals. The compactness type properties of universals are investigatedand it is shown that the class of metric spaces can be characterised as those with a zeroset universal parametrised by a sigma-compact space. It is shown that for a space to have aLindelof-Sigma zero set universal the space must have a sigma-disjoint basis.A study of chain conditions in Ck(X) and Cp(X) is undertaken, giving necessary andsufficient conditions on a space X such that Cp(X) has calibre (kappa,lambda,mu), with a similar resultobtained for the Ck(X) case. Extending known results on compact spaces it is shown that if aspace X is omega-bounded and Ck(X) has the countable chain condition then X must be metric.The classic problem of the productivity of the countable chain condition is investigated inthe Ck setting and it is demonstrated that this property is productive if the underlying spaceis zero-dimensional. Sufficient conditions are given for a space to have a continuous functionuniversal parametrised by a separable space, ccc space or space with calibre omega1.An investigation of the sequential separability of function spaces and products is undertaken. The main results include a complete characterisation of those spaces such that Cp(X)is sequentially separable and a characterisation of those spaces such that Cp(X) is stronglysequentially separable.