This paper reviews several leading approaches for asymptotic modelling of thin layers in elastostatics and wave propagation phenomena. The issues related to applications of the so-called ‘equivalent’ or ‘effective’ boundary conditions and their interpretations are highlighted. Comparative analysis of asymptotic models is performed for a two-dimensional elastostatic case using a novel complex variables-based modelling tool. Its implementation allows for straightforward derivations of higher order boundary conditions for problems with layers of arbitrary sufficiently smooth curvatures. Explicit expressions for the conditions up to the third order are provided. All models are tested using available benchmark solutions and the solutions for the limiting cases of the layer parameters. This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)’.