In the combustion literature, contradictory results on the influence of turbulence on the local thickness of a premixed flame can be found and the present paper aims at contributing to reconcile this issue. First, different measures of local flame thickness in a turbulent flow, e.g. area-weighted and unweighted surface-averaged values of (i) vertical bar del c vertical bar, i.e., the absolute value of 3D gradient of the combustion progress variable c, or (ii) 1/vertical bar del c vertical bar, are studied and analytical relationships/inequalities between them are obtained. Second, the evolution of the different flame thickness measures is explored by numerically evaluating them, as well as various terms in relevant evolution equations derived analytically. To do so, various measures and terms are extracted from DNS data obtained from (i) a highly turbulent, constant-density, dynamically passive, single-reaction wave, (ii) moderately and highly turbulent, single-step-chemistry flames, and (iii) moderately and highly turbulent, complex-chemistry lean methane-air flames. In all those cases, all studied flame thickness measures are reduced during an early stage of premixed turbulent flame development, followed by local flame re-broadening at later stages. Analysis of various terms in the aforementioned evolution equations shows that the initial local flame thinning is controlled by turbulent strain rates. The subsequent local flame re-broadening is controlled by (i) curvature contribution to the stretch rate, which counter-balances the strain rate, (ii) spatial non-uniformities of the normal diffusion contribution to the local displacement-speed vector S(d)n, and (iii) dilatation, which plays an important role in moderately turbulent flames, but a minor role in highly turbulent flames. Moreover, the present study shows that differently defined measures of a local flame thickness can be substantially different. This difference should also be borne in mind when comparing data that indicate local flame thinning with data that indicate local flame broadening. (C) 2019 The Combustion Institute. Published by Elsevier Inc. All rights reserved.