Abstract: Quantization - generally speaking, the process of deforming a `classical’ state space to a `quantum mechanical’ one - is a problem of fundamental importance. It becomes even more challenging when one wants to quantize not just a single space in isolation, but rather to simultaneously quantize a collection of spaces in a manner which is compatible with some key structural maps between them. In fact, this is generally not possible. Nevertheless, in this talk we will describe an approach to this problem which works for a large class of spaces and structural maps which are of interest both mathematically and physically (they generalize moduli spaces of flat connections). The trick is that these spaces can be understood in terms of `colored surfaces' (which we will introduce) which are both quite visual and easy to work with. As one application, we will explain how this allows one to quantize Lie bialgebras (thus, obtaining quantum groups) and to construct certain equivariant quantizations. This talk is based on joint work with Pavol Severa.