This thesis sets itself the goal of investigating function learning approaches. The topic is extremely relevant today, since the abundance of data, jointly with the technological progress which has enhanced computing power, now allow to represent functions of interest with unprecedented prowess starting from available observations. Finance and financial engineering undoubtedly have compelling reasons to exploit such a trend, namely the need to perform computations fast and accurately. Since financial models rely on stochasticity and on complex numerical solutions, it becomes vital both in the academia and for practitioners to deploy methods which overcome the burdens of existing ones. Many issues still remain open. Interpretability and lack of theoretical understanding of existing methods, such as neural networks; data scarcity in some of the financial applications; large data which become intractable and the related curse of dimensionality; the handling of time series and computational issues are among the aspects which deserve further investigation and which will make the object of this work.