  Persistent topological Laplacians constitute a new class of tools in
topological data analysis (TDA), motivated by the necessity to address
challenges encountered in persistent homology when handling complex data. These
Laplacians combines multiscale analysis with topological techniques to
characterize the topological and geometrical features of functions and data.
Their kernels fully retrieve the topological invariants of persistent homology,
while their nonharmonic spectra provide supplementary information, such as the
homotopic shape evolution of data. Persistent topological Laplacians have
demonstrated superior performance over persistent homology in addressing
large-scale protein engineering datasets. In this survey, we offer a
pedagogical review of persistent topological Laplacians formulated on various
mathematical objects, including simplicial complexes, path complexes, flag
complexes, diraphs, hypergraphs, hyperdigraphs, cellular sheaves, as well as
$N$-chain complexes. Alongside fundamental mathematical concepts, we emphasize
the theoretical formulations associated with various persistent topological
Laplacians and illustrate their applications through numerous simple geometric
shapes.
