Spherically expanding flame propagations have been employed to measure flame speeds for H2/CH4/air mixtures over a wide range of H2 fractions (30 %, 50 %, 70 and 100 % hydrogen by volume), at initial temperatures of 303 K and 360 K, and pressures of 0.1, 0.5 and 1.0 MPa. The equivalence ratio (ϕ) was varied from 0.5 to 2.5 for pure hydrogen and from 0.8 to 1.2 for methane/hydrogen mixtures. Experimental laminar burning velocities and Markstein numbers for methane/hydrogen/air mixtures at high pressures, which are crucial for gas turbine applications, are very rare in the literature. Moreover, simulations using three recent chemical kinetic mechanisms (Konnov-2018 detailed reaction, Aramco-2.0-2016 and San Diego Methane detailed mechanism (version 20161214)) were compared against the experimentally derived laminar burning velocities. The maximum laminar burning velocity for 30 % and 50 % H2 occurs at ϕ = 1.1. However, it shifts to ϕ = 1.2 for 70 % H2 and to ϕ = 1.7 for a pure H2 flame. The laminar burning velocities increased with hydrogen fraction and temperature, and decreased with pressure. Unexpected behaviour was recorded for pure H2 flames at low temperature and ϕ = 1.5, 1.7 wherein ul did not decrease when the pressure increased from 0.1 to 0.5 MPa. Although, the measurement uncertainty is large at these conditions, the flame structure analysis showed a minimum decline in the mass fractions of the active species (H, O, and OH) with the rise in the initial pressure. Markstein length (Lb) and Markstein number (Mab and Masr) varied non-monotonically with hydrogen volume fraction, pressure and temperature. There was generally good agreement between simulations and experimentally derived laminar burning velocities, however, for experiments of rich-pure hydrogen at high initial pressures, the level of agreement decreased but remained within the limits of experimental uncertainty.