  It is well established that a wide variety of phenomena in cellular and
molecular biology involve anomalous transport e.g. the statistics for the
motility of cells and molecules are fractional and do not conform to the
archetypes of simple diffusion or ballistic transport. Recent research
demonstrates that anomalous transport is in many cases heterogeneous in both
time and space. Thus single anomalous exponents and single generalized
diffusion coefficients are unable to satisfactorily describe many crucial
phenomena in cellular and molecular biology. We consider advances in the field
of heterogeneous anomalous transport (HAT) highlighting: experimental
techniques (single molecule methods, microscopy, image analysis, fluorescence
correlation spectroscopy, inelastic neutron scattering, and NMR), theoretical
tools for data analysis (robust statistical methods such as first passage
probabilities, survival analysis, different varieties of mean square
displacements, etc), analytic theory and generative theoretical models based on
simulations. Special emphasis is made on high throughput analysis techniques
based on machine learning and neural networks. Furthermore, we consider
anomalous transport in the context of microrheology and the heterogeneous
viscoelasticity of complex fluids. HAT in the wavefronts of reaction-diffusion
systems is also considered since it plays an important role in morphogenesis
and signalling. In addition, we present specific examples from cellular biology
including embryonic cells, leukocytes, cancer cells, bacterial cells, bacterial
biofilms, and eukaryotic microorganisms. Case studies from molecular biology
include DNA, membranes, endosomal transport, endoplasmic reticula, mucins,
globular proteins, and amyloids.
