It is well known and accepted that some natural phenomena exhibit what is known as long-term memory. Since the publication of H.E Hurst’s seminal work in 1951, the study of memory processes has spread to a great range of fields of scientific knowledge, and economics has not been immune to this process. The work of Adelman (Am Econ Rev 55:444–463, 1965) was the starting point of a wide variety of works that have ranged from macroeconomics to finance; the latter are the ones that concentrate the greatest variety of works, mainly as a consequence of the ease of access to data since the computer revolution of financial markets in the late 1990s. It was on that date that physics researchers began to study financial markets and turned the study of memory processes in financial series into one of the most popular topics. In this work, we intend to review the financial literature on memory processes with special interest in the differences between purely economic and physical approaches. We show that this is a very relevant topic for both communities, as the number of papers is still growing. However, as Segovia et al. (Complexity 2022, 2022) noted, both groups publish in journals of their disciplines, and on very limited occasions there are cross-cutting citations. Even academic fields have their own conventions, terminology, and specialized language that may not be familiar to researchers from other disciplines.