Motivated by modelling challenges arising in microfluidics and low-Reynolds-number swimming, we present a new transform approach for solving biharmonic boundary value problems in two-dimensional polygonal and circular domains and show its implementation in various Stokes flow problems. The method is an extension of earlier work by Crowdy & Fokas [Proc. Roy. Soc. A, 460, (2004)] and provides a unified general approach to finding quasi-analytical solutions to a wide range of problems in low-Reynolds-number hydrodynamics and plane elasticity. The new approach leads to fast and accurate schemes for evaluation of the solutions. [Joint work with Darren Crowdy (Imperial).]