Microfluidics has significantly advanced research in environmental science, biomedicine, and chemical analysis. Many natural and synthetic fluids, such as saliva, blood and polymer solution, are non-Newtonian. These fluids can exhibit viscoelastic and shear-thinning properties, which profoundly affect particle behaviour in microchannels. Understanding particle migration in these fluids is essential for accurate and effective particle manipulation. This paper reviews the current viscoelastic microfluidics research. We first explain the fundamental theory of hydrodynamic forces on particles in viscoelastic fluids. Next, we describe particle viscoelastic migration in channels of different geometries and cross-sectional shapes. Subsequently, we review the state of the art of multi-stage and multi-physics integration with viscoelastic microfluidics. We also provide a summary of the applications of viscoelastic microfluidics in manipulating micro and nano bioparticles. Finally, we discuss the limitations of current research and propose future research directions. This review aims to provide insights into viscoelastic microfluidics' further development and applications.