In this paper, thinned concentric circular antenna arrays (CCAAs) with low sidelobe levels and fixed half-power beamwidths are synthesized by using the Slime Mold Algorithm (SMA). SMA is a novel stochastic optimization method inspired by the oscillation mode of the slime mold in nature. Entering the literature as a new optimization technique, SMA is based on a matchless mathematical model that finds the most suitable way to find food in the search space thanks to the negative and positive oscillations of the slime mold. SMA is also used to obtain thinned CCAAs having different ring numbers. It is seen that the results obtained by SMA are very good. Besides, the flexibility of SMA in different values of parameters is a promising feature for the other antenna array synthesis problems. The results obtained by SMA are compared with several meta-heuristic algorithms in the literature. SMA shows a better performance in synthesizing CCAAs than the other compared algorithms.