In this paper, we revisit the view update problem in a relational setting and propose a framework based on the notion of determinacy under constraints. Within such a framework, we characterise when a view mapping is invertible, establishing that this is the case precisely when each database symbol has an exact rewriting in terms of the view symbols under the given constraints, and we provide a general effective criterion to understand whether the changes introduced by a view update can be propagated to the underlying database relations in a unique and unambiguous way. Afterwards, we show how determinacy under constraints can be checked, and rewritings effectively found, in three different relevant scenarios in the absence of view constraints. First, we settle the long-standing open issue of how to solve the view update problem in a multi-relational database with views that are projections of joins of relations, and we do so in a more general setting where views are defined by arbitrary conjunctive queries and database constraints are stratified embedded dependencies. Next, we study a setting based on horizontal decompositions of a single database relation, where views are defined by selections on possibly interpreted attributes (e.g., arithmetic comparisons) in the presence of domain constraints over the database schema. Lastly, we look into another multi-relational database setting, where views are defined in an expressive "Type" Relational Algebra based on the n-ary Description Logic DLR and database constraints are inclusions of expressions in that algebra.