The local turbulent flame speed of an attached flame is not only a function of the local flow and flame conditions, but also of upstream conditions - i.e., it is "non-local" or exhibits "memory". Non-locality adds an additional degree of freedom to the classic problem of a freely propagating flame propagating normally to the time averaged flow. Non-locality occurs due to mean tangential flow along the flame brush, which causes flame wrinkles to translate downstream. As such, the wrinkling of the flame at any given point is not only a function of the local velocity disturbance, but also a superposition of flame surface perturbations from locations upstream at previous times. This causes the correlation length scale of turbulent flame wrinkles to differ from that of the underlying turbulent velocity fluctuations. The objective of this paper is to provide a physical description of the key flame kinematic processes that cause these non-local effects. Two approaches are adopted in this work. First, analytical solutions of the G-equation that explicitly describe the effect of non-locality in the low turbulence limit are developed. Second, numerical computations of the G-equation are performed that demonstrate the role of non-linearity in flame surface kinematics at higher turbulence intensities. Finally, these predictions are shown to be consistent with data from a turbulent Bunsen flame. (C) 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved.