In recent years we have seen the development of efficient and provably correct algorithms for learning weighted automata and closely related function classes such as weighted transducers and weighted context-free grammars. The common denominator of all these algorithms is the so-called spectral method, which gives an efficient and robust way to estimate recursively defined functions from empirical estimations of observable statistics. These algorithms are appealing because of the existence of theoretical guarantees (e.g. they are not susceptible to local minima) and because of their efficiency. However, despite their simplicity and wide applicability to real problems, their impact in NLP applications is still moderate. One of the goals of this tutorial is to remedy this situation.The contents that will be presented in this tutorial will offer a complementary perspective with respect to previous tutorials on spectral methods presented at ICML-2012, ICML-2013 and NAACL-2013. Rather than using the language of graphical models and signal processing, we tell the story from the perspective of formal languages and automata theory (without assuming a background in formal algebraic methods). Our presentation highlights the common intuitions lying behind different spectral algorithms by presenting them in a unified framework based on the concepts of low-rank factorizations and completions of Hankel matrices. In addition, we provide an interpretation of the method in terms of forward and backward recursions for automata and grammars. This provides extra intuitions about the method and stresses the importance of matrix factorization for learning automata and grammars. We believe that this complementary perspective might be appealing for an NLP audience and serve to put spectral learning in a wider and, perhaps for some, more familiar context. Our hope is that this will broaden the understanding of these methods by the NLP community and empower many researchers to apply these techniques to novel problems.The content of the tutorial will be divided into four blocks of 45 minutes each, as follows. The first block will introduce the basic definitions of weighted automata and Hankel matrices, and present a key connection between the fundamental theorem of weighted automata and learning. In the second block we will discuss the case of probabilistic automata in detail, touching upon all aspects from the underlying theory to the tricks required to achieve accurate and scalable learning algorithms. The third block will present extensions to related models, including sequence tagging models, finite-state transducers and weighted context-free grammars. The last block will describe a general framework for using spectral techniques in more general situations where a matrix completion pre-processing step is required; several applications of this approach will be described.