An analysis is made of the laminar heat transfer characteristics of an array of longitudinal fins with an adiabatic shroud situated adjacent to the fin tips. The analysis involves the solution of the velocity field in the inter-fin space and in the shroud clearance gap beyond the tips, after which the governing energy equations for the fluid and the fins are. solved simultaneously. Solutions are obtained for representative values of dimensionless parameters which describe the system geometry and the fin conductance. For the fin, the results show that the heat loss is a minimum adjacent to the base and increases along the fin in the direction of the tip. The maximum fin heat loss occurs either at the tip or intermediate between the base and the tip, depending on whether or not there is clearance between the fin tips and the shroud. The calculated heat transfer coefficients vary along the fin and, in some cases, take on negative values. Heat loss variations are also encountered along the base surface, the extent of which depends on the fin conductance, the inter-fin spacing, and the extent of the clearance gap. With regard to the overall heat loss, the fin is, on a unit area basis, a more efficient transfer surface than is the base. The results demonstrate that the conventional uniform heat transfer coefficient model is inapplicable to shrouded fin arrays.