In this paper a methodology and an application example are presented aiming to show how Artificial Neural Networks (ANNs) can be used to model manufacturing processes when mathematical models are missing or are not applicable e.g. due to the micro- & nano-scaling, due to non-conventional processes, etc. Besides the ANNs methodology, the results of a Software System developed will be presented, which was used to create ANNs models for micro & nano manufacturing processes. More specifically results of a specific application example will be presented, concerning the modeling of extrusion processes for polymeric micro-tubes. ANNs models are capable for modeling manufacturing processes as far as adequate experimental and/or historical data of processes' inputs & outputs are available for their training. The POLYTUBES ANNs models have been trained and tested with experimental data records of process' inputs and outputs concerning a micro-extrusion process of polymeric micro-tubes for several materials such as: COC, PC, PET, PETG, PP and PVDF. The main ANN model of the extrusion application example has 3 inputs and 9 outputs. The inputs are: tube's inner & outer diameters, and the material density. The model outputs are 9 process parameters, which correspond to the specific inputs e.g. process temperature, die inner & outer diameters, extrusion pressure, draw speed etc. The training of the ANN model was completed, when the errors for the model's outputs, which expressed the difference between the training target values and the ANNs outputs, were minimized to acceptable levels. After the training, the micro-extrusion ANN is capable to simulate the process and can be used to calculate model's outputs, which are the process parameters for any new set of inputs. By this way a satisfactory functional approximation of the whole process is achieved. This research work has been supported by the EU FP7 NMP project POLYTUBES.