The 1-1/2 fluid is a conformal solution in which one species has a size parameter (diameter) of zero. This "point-particle" species nevertheless interacts with the other component of the mixture, as the collision diameter of a point particle and a finite particle is nonzero. A great simplifying feature of this model mixture is that the point particles do not interact with each other. For hard repulsive potentials, the properties of a 1-1/2 fluid can be obtained exactly in terms of the properties of the pure fluid obtained upon removal of all the point particles. For other potentials, the properties of the 1-1/2 fluid can be obtained only approximately. We develop two approaches to the description of the 1-1/2 fluid, both based on the methods of diagrammatic expansion and topological reduction. The first approach is an extended virial treatment, in which the free energy is expanded in the density p(1) of the full-sized species, keeping to all orders terms in the density p(2) of the point particles. A complementary approach takes the pure full-sized fluid as a reference, keeping all terms to p(1) while expanding in p(2). Monte Carlo simulation is used to show that, properly formulated, an expansion containing only terms first order in p(2) is capable of describing 1-1/2 fluid properties over a very broad range of conditions.