We formulate a simple theoretical model that permits one to investigate surface-tension-driven flows with complex interface geometry. The model consists of a Hele-Shaw cell filled with two different fluids and subjected to a unidirectional temperature gradient. The shape of the interface that separates the fluids can be arbitrarily complex. If the contact line is pinned, i.e. unable to move, the problem of calculating the flow in both fluids is governed by a linear set of equations containing the characteristic aspect ratio and the viscosity ratio as the only input parameters. Analytical solutions, derived for a linear interface and for a circular drop, demonstrate that for large aspect ratio the flow field splits into a potential core flow and a thermocapillary boundary layer which acts as a source for the core. An asymptotic theory is developed for this limit which reduces the mathematical problem to a Laplace equation with Dirichlet boundary conditions. This problem can be efficiently solved utilizing a boundary element method. It is found that the thermocapillary flow in non-circular drops has a highly non-trivial streamline topology. After releasing the assumption of a pinned interface, a linear stability analysis is carried out for the interface under both transverse and longitudinal temperature gradients. For a semi-infinite fluid bounded by a freely movable surface long-wavelength instability due to the temperature gradient across the surface is predicted. The mechanism of this instability is closely related to the long-wave instability in surface-tension-driven Bénard convection. A linear interface heated from the side is found to be linearly stable. The possibility of experimental verification of the predictions is briefly discussed.