Detailed kinetic-transport models were used to explore thermodynamic and kinetic barriers in the non-oxidative conversion of CH4 via homogeneous and homogeneous-heterogeneous pathways and the effects of continuous hydrogen removal and of catalytic sites on attainable yields of useful C2-C10 products. The homogeneous kinetic model combines separately developed models for low-conversion pyrolysis and for chain growth to form large aromatics and carbon. The H2 formed in the reaction decreases CH4 pyrolysis rates and equilibrium conversions and it favors the formation of lighter products. The removal of H2 along tubular reactors with permeable walls increases reaction rates and equilibrium CH4 conversions. C2-C10 yields reach values greater than 90 percent at intermediate values of dimensionless transport rates (delta=1-10), defined as the ratio hydrogen transport and methane conversion rates. Homogeneous reactions require impractical residence times, even with H2 removal, because of slow initiation and chain transfer rates. The introduction of heterogeneous chain initiation pathways using surface sites that form methyl radicals eliminates the induction period without influencing the homogeneous product distribution. Methane conversion, however, occurs predominately in the chain transfer regime, within which individual transfer steps and the formation of C2 intermediates become limited by thermodynamic constraints. Catalytic sites alone cannot overcome these constraints. Catalytic membrane reactors with continuous H2 removal remove these thermodynamic obstacles and decrease the required residence time. Reaction rates become limited by homogeneous reactions of C2 products to form C6 aromatics. Higher delta values lead to subsequent conversion of the desired C2-C10 products to larger polynuclear aromatics. We conclude that catalytic methane pyrolysis at the low temperatures required for restricted chain growth and the elimination of thermodynamics constraints via continuous hydrogen removal provide a practical path for the direct conversion of methane to higher hydrocarbons. The rigorous design criteria developed are being implemented using shape-selective bifunctional pyrolysis catalysts and perovskite membrane films in a parallel experimental effort.