Today’s AI technologies, such as Watson and Siri, are impressive yet still confined to a single domain or task. Imagine how truly intelligent systems---systems that actually understand their world---could change our world. A successful research plan for understanding intelligence includes two key domains: the domain of the physical world and the domain of human agents and their interactions. First, understanding intelligence requires scene, object and action recognition; second, it requires non-verbal social perception (NVSP). As an example of research in the first domain, I will describe work at the joint IIT-MIT Laboratory for Computational and Statistical Learning over the last 2 years developing a theory of visual cortex and of deep learning architectures of the convolutional type. I will describe the theoretical consequences of a simple assumption: the main computational goal of the feedforward path in the ventral stream – from V1, V2, V4 and to IT – is to discount image transformations, after learning them during development. A basic neural operation consists of dot products between input vectors and synaptic weights – which can be modified by learning. I will outline theorems showing that a multi-layer hierarchical architecture of dot-product modules can learn in an unsupervised way geometric transformations of images and then achieve the dual goals of invariance to global affine transformations and of robustness to deformations. These architectures can achieve invariance to transformations of a new object: from the point of view of machine learning they show how to learn in an unsupervised way representations that may reduce considerably the sample complexity of supervised learning.