Bayes linear methodology provides a quantitative structure for expressing our beliefs and systematic methods for revising these beliefs given observational data. Particular emphasis is placed upon interpretation of and diagnostics for the specification. The approach is similar in spirit to the standard Bayes analysis, but is constructed so as to avoid much of the burden of specification and computation of the full Bayes case. This report is the first of a series describing Bayes linear methods. In this document, we introduce some of the basic machinery of the theory. Examples, computational issues, detailed derivations of results and approaches to belief elicitation will be addressed in related reports.