Propagating surface evolution in isotropic turbulence is studied using velocity fields generated by direct numerical simulations. The statistics of tangential strain rate, fluid velocity, characteristic curvature and area-following propagating surface elements are investigated. The one-time statistics of strain rate and fluid velocity pass monotonically from Lagrangian value at low propagation speeds to Eulerian values at high speeds. The strain-rate statistics start deviating significantly from the Lagrangian values only for propagating velocities greater than the Kolmogorov velocity scale vη, whereas, with fluid-velocity statistics the deviation occurs only for velocities greater than the turbulence intensity u′. The average strain rate experienced by a propagating surface decreases from a positive value to near zero with increasing propagation velocity. The autocorrelation function and frequency spectrum of fluid velocity and strain rate scale as expected in the limits of small and large propagation velocities. It is also found that for the range of propagation velocities considered, an initially plane surface element in turbulence develops a cusp in finite time with probability nearly one. The evolution of curvature is studied using the concept of hitting time. Initially plane propagating surfaces end up being almost cylindrical in shape. Highly curved surface elements are associated with negative strain rates and small surface areas.