In this dissertation, a lubrication-type model is developed that describes liquid flow and heat transfer in different groove structures in micro heat pipes under negligible gravity and small capillary number. Determination of the vapor-liquid interface shape in the grooves requires coupling of fluid flow and heat transfer. We examined the viscous flow and the shape of the interface of the heat pipe and found that in the adiabatic region, the flow can be controlled by changing the shape of the cross-section. We have computed the maximum flow rate as a function of the geometry. This study is of interest for the design of heat pipes since the efficiency of heat removal depends on the amount of flow that can be sustained by capillary forces in the adiabatic region. When evaporation is taken into account, the flow rate changes along the channel due to evaporative mass loss, which also affects the shape of the interface. The thermocapillary effect is studied under conditions of negligible evaporation. In particular, the extent of wetted area changes along the axis of the wedge as a function of evaporation is studied. Detailed analysis of liquid flow in the triangular grooves and comparisons with previous models and experiments are discussed. Rectangular and advanced capillary cross-sections are also considered in the study. Practical applications of our results are discussed in relation to the design and optimization of micro heat pipes. Numerical boundary integral approach to the study of interface in the region where lubrication theory breaks down is also described.