Naturally formed nervous systems such as a brain are extremely complex, and their precisely wired connectivity appears to be behind their rich activity repertoire and astonishing computational capabilities. However, the same neurons dissociated and cultured on glass coverslips show simple activity patterns characterized by the synchronous \"all-or-none\" firing of the network\'s population. To uncover the secrets of neuronal connectivity and its role in governing activity, I have been working for the last five years in new experimental platforms to describe connectivity in living neuronal networks. In the first part of the talk I will introduce a graph-theoretical approach that, combined with measurements on neural activity, provides relevant statistical information of the neural connectivity. Among other quantities, we can assess the average connectivity of the network or the distribution of connections. The latter appears to be Gaussian in two-dimensional neuronal cultures, very different from the small-world or power law distributions present in natural nervous systems. In the second part of the talk I will introduce new experimental strategies currently under development in my group. These strategies are focused on the confinement of neurons and connections in predefined patterns to control the network\'s circuitry. These \"engineered neural networks\" appear to have a much more complex connectivity and richer activity patterns than standard cultures, and are excellent model systems for the advancement of network-theory concepts and neuronal computation.