Wireless sensor networks have emerged as versatile tools to monitor the evolution of physical fields over large areas by means of self-powered and low-cost sensing devices. Although their development was originally motivated by military applications, they are nowadays used in numerous civilian application areas. In many sensor network scenarios, including the ones studied in this thesis, the goal of data acquisition by means of sensor nodes is to reproduce at a base station the observed field under a fidelity constraint, using limited communication resources. Due to the scarcity of the energy resources, the physical nature of the observed fields, the constraints on spatial sampling, the high cost of inter-node communication and the reliance on a shared communication medium, wireless sensor networks raise challenging questions in the areas of multidimensional sampling, multiterminal source coding and network communications. In this thesis we address the issue of how to efficiently represent the data collected by the sensor nodes. This data has some inherent structure that is determined by the laws of physics. Since the design of efficient sampling geometries and source coding schemes requires a thorough understanding of the physical properties of the observed phenomenon, the first part of our work is devoted to setting up a mathematical model for physical fields that is amenable to the tools of information theory while applying to most physical phenomena of interest. The question regarding the sufficiency of a discrete representation of an analog field is addressed in two steps. First, we prove a multidimensional sampling theorem for homogeneous random fields with compactly supported spectral measures. While, under appropriate conditions, a discrete-space and -time representation of the analog field is sufficient, discretizing the field's amplitude, also referred to as source coding, implies an unavoidable loss of information, which may be expressed in terms of a rate distortion function. In the second step we study various source coding schemes, differing by the amount of required inter-node communication and the complexity of the involved maps. We derive lower and upper bounds for the rate distortion function of the multiterminal source coding scheme, which is of particular interest in sensor network engineering as it makes use of the spatio-temporal structure of the collected data without requiring inter-node communication. In the second part of the thesis we study some applications of sensor networks. In particular, we consider the acquisition of acoustic waves, the monitoring of temperature distributions and the compression of data sequences generated by random walks. For the setup of sound field acquisition, we show that, under the assumption of spectral whiteness of the sound field, the afore-mentioned bounds coincide, thus establishing the multiterminal rate distortion function for this setup.