The general question, crucial to an understanding of the internal structure of the nucleon, of how to split the total angular momentum of a photon or gluon into spin and orbital contributions is one of the most important and interesting challenges faced by gauge theories like Quantum Electrodynamics and Quantum Chromodynamics. This is particularly challenging since all QED textbooks state that such an splitting cannot be done for a photon (and a fortiori for a gluon) in a gauge-invariant way, yet experimentalists around the world are engaged in measuring what they believe is the gluon spin! This question has been a subject of intense debate and controversy, ever since, in 2008, it was claimed that such a gauge-invariant split was, in fact, possible. We explain in what sense this claim is true and how it turns out that one of the main problems is that such a decomposition is not unique and therefore raises the question of what is the most natural or physical choice. The essential requirement of measurability does not solve the ambiguities and leads us to the conclusion that the choice of a particular decomposition is essentially a matter of taste and convenience. In this review, we provide a pedagogical introduction to the question of angular momentum decomposition in a gauge theory, present the main relevant decompositions and discuss in detail several aspects of the controversies regarding the question of gauge invariance, frame dependence, uniqueness and measurability. We stress the physical implications of the recent developments and collect into a separate section all the sum rules and relations which we think experimentally relevant . We hope that such a review will make the matter amenable to a broader community and will help to clarify the present situation.C.Lorce@ulg.ac.be 30 1. The Stueckelberg symmetry 30 2. Towards a more refined classification 31 3. Origin and geometrical interpretation of the Stueckelberg symmetry 33 4. Measurability and the controversy about Stueckelberg symmetry 35 D. The Lorentz transformation properties 38 1. The standard approach 38 2. Critique of the standard approach 39 3. Lorentz transformation law of the pure-gauge and physical fields 40 V. The proton spin decomposition 41 A. The QCD energy-momentum and covariant angular momentum tensors 42 B. Decompositions of the proton momentum and the proton spin 44 1. The canonical decompositions 44 2. The kinetic decompositions 45 3. The master decomposition 46 C. Non-abelian Stueckelberg and Lorentz transformations 48 Bel,z 74 2. Lattice calculation of J q Bel,T 76 3. Evaluation of L q Ji,z in a longitudinally polarized nucleon, from GPDs 77 4. Evaluation of L q Ji,z in a longitudinally polarized nucleon, from GTMDs 78 B. Expressions for the canonical version of L 79 C. The orbital angular momentum in quark models 79 D. The phase-space distribution of angular momentum 82 VIII. Qualitative summary and experimental implications 84 A. Gauge invariance and measurability 84 B. Two kinds of decompositions 85 C. Sum rules vs. relations 86 1...