XPath is the standard language for navigating XML documents and returning a set of matching nodes. We present a sound and complete decision procedure for containment of XPath queries as well as other related XPath decision problems such as satisfiability, equivalence, overlap and coverage. The considered XPath fragment covers most of the language features used in practice. Specifically, we propose a unifying logic for XML, namely the alternation-free modal µ-calculus with converse. We show how to translate major XML concepts such as XPath and regular XML types (including DTDs) into this logic. Based on these embeddings, we show how XPath decision problems, in the presence or the absence of XML types, can be solved using a decision procedure for µ-calculus satisfiability. We provide a complexity analysis together with practical experiments of our system that illustrate the efficiency of the approach for realistic scenarios.