It is well known that near sharp edges or boundary frames the slope of minimal surfaces become large or even infinity. On the other hand, in certain biological membranes and material interfaces the slope tends to be bounded since their configurations are not highly curved due to limited energy. Imposing a bounded slope constraint may create a free boundary when the slope threshold is attained, giving rise to Lagrange multipliers which have a physical meaning. Although the slope is a local property, for grain boundaries and interfaces a nonlocal gradient may be useful for modelling certain phenomena as approximation of the local models. In joint works with Lisa Santos and Pedro Campos, we shall discuss the well-posedness of the variational approach to local and nonlocal minimal and capillary nonparametric surfaces with bounded slopes.