This paper examines attribute dependencies in data that involve grades, such as a grade to which an object is red or a grade to which two objects are similar. We thus extend the classical agenda by allowing graded, or fuzzy, attributes instead of Boolean attributes in case of attribute implications, and allowing approximate match based on degrees of similarity instead of exact match in case of functional dependencies. In a sense, we move from bivalence, inherently present in the now-available theories of dependencies, to a more flexible setting that involves grades. Such a shift has far-reaching consequences. We argue that a reasonable theory of dependencies may be developed by making use of mathematical fuzzy logic. Namely, the theory of dependencies is then based on a solid logic calculus the same way the classical dependencies are based on classical logic. For instance, rather than handling degrees of similarity in an ad hoc manner, we consistently treat them as truth values, the same way as true (match) and false (mismatch) are treated in classical theories. In addition, several notions intuitively embraced in the presence of grades, such as a degree of validity of a particular dependence or a degree of entailment, naturally emerge and receive a conceptually clean treatment in the presented approach. In the paper, we discuss motivations, provide basic notions of syntax and semantics, and develop basic results which include entailment of dependencies, associated closure structures, a logic of dependencies with two versions of completeness theorem, results and algorithms regarding complete non-redundant sets of dependencies, relationship to and a possible reductionist interface to classical dependencies, and relationship to functional dependencies over domains with similarity.