Live cells and intracellular structures and components are in constant motion. Their movement in space is always associated with complex auto-wave hydrodynamics of protoplasm. Corresponding mathematical models were constructed for the giant cells of green Charcean algae and the plasmodium of slime mold Physarum. The microflows of protoplasm in these giant cells are supported by the work of molecular motors (MM) operating on the basis of actin-myosin interactions. However, in the protoplasm of cells of smaller sizes, such as neurons, continuous directional transport of organelles also takes place. It is sustained with the help of MM, operating on the basis of the interaction of such protein macromolecules as kinesin and myosin V. In this regard, a general picture of transport phenomena in neurons is presented. It should be emphasized that both the protoplasmic microflows in live cells and the dynamics MM were studied to a large extent using a variety of laser and optical methods: Doppler microscopy, micro video imaging, laser tweezers and magnetic traps of different designs, and methods of recording various optical markers associated with moving fragments of MM. In this paper, we shall overview the optical techniques for studying both the protoplasmic streaming and the organelle transport in neurons. A detailed discription of kinesin functioning will be given and relevant new mathematical models of its molecular dynamics will be proposed.