Rigid or flexible particles suspended in viscoelastic fluids are ubiquitous in the food industry (e.g. pastes), industrial molding applications (all composites and 3-D printed parts), the energy industry (e.g. fracking fluids), and biological fluids (i.e. swimming of bacteria in mucous). The mathematics of the description of these suspensions is in its infancy. However, while the mathematics of this subject is subtle a major breakthrough in this area has been the development of computational simulations of such viscoelastic suspensions, with particle level resolution, such that predictions can be made and tested at all volume fraction loadings. I describe the use of an Immersed Boundary methodology that allows the simulation of hundreds of particles in elastic fluids, with particle level flow and stress field resolution. This simulation capability is unique and overcomes the major hurdle in understanding the physics of these suspensions – which in many cases are simply qualitatively different than that of Newtonian suspensions. The simplest flows of such suspensions are not understood at a fundamental level, primarily because the collective behavior of particles in an elastic liquid has no foundation – this will change dramatically in the next few years. I will describe three foundational problems that have now been analyzed using this new computational method – including fracking fluid design and swimming in mucous.