The Internet of Things (IoT) has ushered in an age where low-power sensors generate data which are communicated to a back-end cloud for massive data computation tasks. From the hardware perspective this implies co-existence of several power-efficient sub-systems working harmoniously at the sensor nodes capable of communication and high-speed processors in the cloud back-end. The package-board system-level design plays a crucial role in determining the performance of such low-power sensors and high-speed computing and communication systems. Although there exist several commercial solutions for electromagnetic and circuit analysis and verification, problem diagnosis and design tools are lacking leading to longer design cycles and non-optimal system designs. This work aims at developing methodologies for faster analysis, sensitivity based diagnosis and multi-objective design towards signal integrity and power integrity of such package-board system layouts. The first part of this work aims at developing a methodology to enable faster and more exhaustive design space analysis. Electromagnetic analysis of packages and boards can be performed in time domain, resulting in metrics like eye-height/width and in frequency domain resulting in metrics like s-parameters and z-parameters. The generation of eye-height/width at higher bit error rates require longer bit sequences in time domain circuit simulation, which is compute-time intensive. This work explores learning based modelling techniques that rapidly map relevant frequency domain metrics like differential insertion-loss and cross-talk, to eye-height/width therefore facilitating a full-factorial design space sweep. Numerical results performed with artificial neural network as well as least square support vector machine on SATA 3.0 and PCIe Gen 3 interfaces generate less than 2% average error with order of magnitude speed-up in eye-height/width computation. Accurate power distribution network design is crucial for low-power sensors as well as a cloud sever boards that require multiple power level supplies. Achieving target power-ground noise levels for low power complex power distribution networks require several design and analysis cycles. Although various classes of analysis tools, 2.5D and 3D, are commercially available, the presence of design tools is limited. In the second part of the thesis, a frequency domain mesh-based sensitivity formulation for DC and AC impedance (z-parameters) is proposed. This formulation enables diagnosis of layout for maximum impact in achieving target specifications. This sensitivity information is also used for linear approximation of impedance profile updates for small mesh variations, enabling faster analysis. To enable designing of power delivery networks for achieving target impedance, a mesh-based decoupling capacitor sensitivity formulation is presented. Such an analytical gradient is used in gradient based optimization techniques to achieve an optimal set of decoupling capacitors with appropriate values and placement information in package/boards, for a given target impedance profile. Gradient based techniques are far less expensive than the state of the art evolutionary optimization techniques used presently for a decoupling capacitor network design. In the last part of this work, the functional similarities between package-board design and radio frequency imaging are explored. Qualitative inverse-solution methods common to the radio frequency imaging community, like Tikhonov regularization and Landweber methods are applied to solve multi-objective, multi-variable signal integrity package design problems. Consequently a novel Hierarchical Search Linear Back Projection algorithm is developed for an efficient solution in the design space using piecewise linear approximations. The presented algorithm is demonstrated to converge to the desired signal integrity specifications with minimum full wave 3D solve iterations.