Developing methods to process irregularly structured data is crucial in applications like gene-regulatory, brain, power, and socioeconomic networks. Graphs have been the go-to algebraic tool for modeling the structure via nodes and edges capturing their interactions, leading to the establishment of the fields of graph signal processing (GSP) and graph machine learning (GML). Key graph-aware methods include Fourier transform, filtering, sampling, as well as topology identification and spatiotemporal processing. Although versatile, graphs can model only pairwise dependencies in the data. To this end, topological structures such as simplicial and cell complexes have emerged as algebraic representations for more intricate structure modeling in data-driven systems, fueling the rapid development of novel topological-based processing and learning methods. This paper first presents the core principles of topological signal processing through the Hodge theory, a framework instrumental in propelling the field forward thanks to principled connections with GSP-GML. It then outlines advances in topological signal representation, filtering, and sampling, as well as inferring topological structures from data, processing spatiotemporal topological signals, and connections with topological machine learning. The impact of topological signal processing and learning is finally highlighted in applications dealing with flow data over networks, geometric processing, statistical ranking, biology, and semantic communication.