This text studies two-dimensional magnetohydrodynamic (MHD) transport processes in incompressible non-Newtonian fluids using similarity transformations and the Homotopy Analysis Method (HAM). It systematically brings together introductory concepts and two primary mathematical boundary-layer models:• Model 1 (Casson Fluid): Explores steady flow over a heated stretching sheet under a transverse magnetic field. It incorporates thermal radiation, chemical reactions, heat generation, and cross-diffusion (Soret and Dufour effects). Results show that increasing the magnetic parameter (M) or Casson parameter (γ) retards fluid velocity, while thermal/solutal buoyancy acts to enhance it.• Model 2 (Viscoelastic Fluid): Considers steady flow over a stretching or shrinking plate near a stagnation point. It features Brownian motion and thermophoresis particle-slip mechanisms alongside radiation and a convective boundary condition. Elastic (K) and magnetic effects are found to diminish velocity, whereas stagnation parameter (s) broadens the profile.