Collision avoidance for fixed-wing unmanned aerial vehicles (UAVs), using a novel methodology of avoidance map, is addressed in this work. Avoidance map is defined as a mapping in the control input space of a pair of UAVs, which partitions the control space into avoidance and collision regions, respectively. The choice of maneuvers to execute successful avoidance can be effectively obtained by using the avoidance map. The duration of control application is determined by a simple method that is time optimal and computationally efficient. Different versions of the map portraying the evolution of the concept are also given. Precision control, defined as a gradual reduction in control, is discussed to reduce the control input demand for the entire duration of avoidance. The concept is extended to multiple UAVs, and the additional requirement of the UAVs to return to nominal trajectory after avoidance, using Dubins paths, is also addressed. Finally, simulation results using realistic UAV models are considered, and some needed modifications to the basic avoidance strategy, made necessary by realistic maneuver models, are discussed. Several illustrative simulation results are given to support the applicability of the proposed method.