People make decisions together every day: they participate in elections, boards of companies, juries of courts, and social activities amongst friends. Accurately capturing such fine-grained human behaviour is essential, not only for directly improving the choices made by people, but also for developing AI (multiagent) systems that work efficiently and safely with collective decisions. This thesis lies in the field of computational social choice, a sub-area of economics and computer science concerned with the analysis of mechanisms that groups use to make decisions, i.e., aggregation mechanisms. Much research to date has been resting on a rather stringent hypothesis—namely, that people are able to form concrete opinions about all issues at stake in a given decision problem (such opinions are called complete). This thesis focuses on collective decisions where the agents may hold intrinsically incomplete opinions, meaning that they may lack an opinion about some of the issues with which they are presented. We investigate aggregation mechanisms with incomplete individual opinions in three contexts, where we care about (i) reaching a consensus between the diverse opinions of the group members for issues that are subjective in nature (like in a travel website that aims at finding the best accommodation based on the reviews of its users); (ii) avoiding skewed outcomes caused by the strategic behaviour of selfish agents (like in election problems); (iii) discovering the ground truth with respect to a number of objective issues (like in crowdsourcing experiments).