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This thesis aims at investigating size dependence of properties of nanostructures from the point of view of a general scaling theory that smoothly connects properties of the bulk to that of nanostructures. Two different examples of a ``static'' and a ``dynamic'' property are considered in this study. The first example studied is of size dependence of coefficient of thermal expansion (CTE) which a static property of nanostructures. The CTE of nanobars and nanoslabs is studied using equilibrium molecular dynamics and dynamical matrix formulation in an electrically insulating medium. It is found that the fractional change in CTE from the bulk value scales inversely with the size of the nanostructures, thus, showing a simple description in terms of a scaling theory. In the second part, electron transport in carbon nanotube field effect transistors (CNTFETs) is studied using Landauer formalism. A CNTFET involves transport through a 1-d ballistic carbon nanotube channel with Schottky barriers (SB) at contacts which determines the transport characteristics. The CNT is modeled as a 1-d semiconductor having only two bands separated by an energy gap which depends inversely on tube diameter. After the contact is made, a self-consistent potential appears due to charge transfer between CNT and metal, which is calculated by solving Poisson equation. The electron transmission across the barriers is calculated using WKB approximation. Current and conductance are calculated using Landauer-Buttiker formula. Diameter dependence of properties like, conductance, threshold voltage, VON, etc. is calculated. It is found that there is no simple scaling for a property for small values of diameter. The scaling form is, however, found to be valid for larger diameters. Also, other calculated device characteristics are in close agreement with experiments. The model presented in this thesis is the first detailed study illustrating the applicability of the scaling approach to the properties of nanostructures. The static properties show scaling behavior, while ``dynamic'' properties derived from electronic response do not.