The analysis of different physical systems and mathematical devices depends on the utilization of various types of algebraic quantities involved in the description of geometrical aspects of the phenomena and states which occur. Certain types of quantities are commonly identified as scalars, vectors and tensors. Among these, the tensors are quite complicated geometric structures as they involve not only magnitude and direction, but also have a dependence on orientation. The most familiar tensors in physical importance are internal stress in a solid and viscous stress in fluid. In both cases, the magnitude and direction of stress as a force per unit area depend on the orientation of the area on which it is acting.

The purpose of the present manuscript is to discuss a nice and lucid characterization of tensor and their basic features. Moreover, the manuscript is intended to serve the purpose of familiarity with recent developments in the tensor theory and its applications in various fields of sciences and Bio-sciences. We have just reviewed and combined the results on applications of tensors, which we feel useful in the recent developments. The manuscript begins with tensor primer and consists of few applications in elasticity including illustrations of stress and deformation tensors of elastic bodies, electro-dynamics with Maxwell's tensor and finally includes brief notions of diffusion tensors used in MRI and strain Green's tensors applied in the study of seismology.