Random walks represent fundamental search strategies for both animal and robots, especially when there are no environmental cues that can drive motion, or when the cognitive abilities of the searching agent do not support complex localisation and mapping behaviours. In swarm robotics, random walks are basic building blocks for the emergent collective behaviour. However, there has been limited account for the correct parameterisation to be used in different search scenarios, and the relationship between search efficiency and information transfer within the swarm has been often overlooked. In this talk, I will present a systematic analysis of different mobility patterns for robot swarms, including Brownian motion and Levy walks, which are evaluated for the ability to find an immobile target and to spread widely information about the target within the swarm. Then, we present two case studies in which the mobility pattern plays a major role: a consensus decision problem and a coverage and mapping problem. We show that the choice of the correct parameters for the random walk is key to the attainment of efficient swarm robotics solutions.