## 1 Introduction
While interfering counterpropagating beams create the first regular pattern, randomness is added by superimposing a second optical lattice, providing a viable strategy to control the degree of disorder [23; 24; 25]. An alternative is given by the speckle optical potential, generated by the interference of waves with different phases and amplitudes but with the same frequency [26; 27]. The resulting wave varies randomly in space and can be used to study the transport properties of ultracold atoms in the presence of static external disorder [28; 29; 30]. In order to tackle down the crucial aspects of the dirty bosons problem, the relevant theoretical tools can be divided in two different subsets. At first, one can adopt the point of view of Bogoliubov theory, treating disorder in a perturbative way, similarly to quantum and thermal fluctuations in the pure system [31; 32]. Since the first seminal work on this topic [33; 34; 35], this approach has led us to a better understanding of the role played by disorder, no matter how small, in condensed bosonic systems. The additional depletion to condensate and superfluid was computed in [33], while, in [34], the authors investigated the modifications induced by a random external potential to the sound propagation. At the same time, critical properties are investigated by focusing, for instance, on the condensation temperature shift driven by the interplay of disorder and interaction [36; 37]. A non-perturbative approach paves the way to the exploration of the superfluid-glass (quantum) transition in bosonic ensembles. Moving from the seminal renormalization-group analysis for one-dimensional systems [7; 38], different strategies and methods have been used to understand the phase diagram of dirty bosons in three and lower dimensions, such as the random-phase approximation [39], stochastic field-theory techniques and the replica formalism [40; 41; 42; 43]. The ambition of this review is obviously more limited, since it is not possible to handle all the topics we have mentioned during this introduction in the main text. Nevertheless, we aim to present, in a clear and pedagogical way, the crucial ingredients of a field-theory analysis of superfluid bosons. Throughout the paper, we are going to adopt the functional integration framework, which is flexible enough to enable the implementation of different approximation schemes [44]. Our discussion is organized in the following way: in Section 2, we specify how one can build up a quantum field theory for superfluid bosons subject to a random confinement. In particular, one needs to clarify the way in which disorder is introduced and characterized. In Section 3, we compute, up to the Gaussian level of approximation, the additional contribution to the condensate and the superfluid component due to the presence of a quenched disordered potential. Moving to Section 4, we present the core ideas at the foundation of the replica formalism, which allows for generalizing our analysis to more complex disorder realization and does not rely, in principle, upon the assumption of a perturbatively small disorder. Within this scheme, we show how to compute the meaningful correlation functions and describe the superfluid response of the system in Section 5. Further comments and future perspectives are addressed in the Conclusions section.