  Topological Data Analysis (TDA) is a discipline that applies algebraic
topology techniques to analyze complex, multi-dimensional data. Although it is
a relatively new field, TDA has been widely and successfully applied across
various domains, such as medicine, materials science, and biology. This survey
provides an overview of the state of the art of TDA within a dynamic and
promising application area: industrial manufacturing and production,
particularly within the Industry 4.0 context. We have conducted a rigorous and
reproducible literature search focusing on TDA applications in industrial
production and manufacturing settings. The identified works are categorized
based on their application areas within the manufacturing process and the types
of input data. We highlight the principal advantages of TDA tools in this
context, address the challenges encountered and the future potential of the
field. Furthermore, we identify TDA methods that are currently underexploited
in specific industrial areas and discuss how their application could be
beneficial, with the aim of stimulating further research in this field. This
work seeks to bridge the theoretical advancements in TDA with the practical
needs of industrial production. Our goal is to serve as a guide for
practitioners and researchers applying TDA in industrial production and
manufacturing systems. We advocate for the untapped potential of TDA in this
domain and encourage continued exploration and research.
