It is generally believed that random search processes based on scalefree, Lévy stable jump length distributions (Lévy flights) optimize the search for sparse targets. Here we show that this popular search advantage is less universal than commonly assumed. We study the efficiency of a minimalist search model based on Lévy flights in the absence and presence of an external drift (underwater current, atmospheric wind, a preference of the walker owing to prior experience, or a general bias in an abstract search space) based on two different optimization criteria with respect to minimal search time and search reliability (cumulative arrival probability). Although Lévy flights turn out to be efficient search processes when the target is far from the starting point, or when relative to the starting point the target is upstream, we show that for close targets and for downstream target positioning regular Brownian motion turns out to be the advantageous search strategy. Contrary to claims that Lévy flights with a critical exponent α = 1 are optimal for the search of sparse targets in different settings, based on our optimization parameters the optimal α may range in the entire interval (1, 2) and especially include Brownian motion as the overall most efficient search strategy.search optimization | stochastic processes | Lévy foraging hypothesis H ow do you find the proverbial needle in the haystack or an enemy submarine in the vast expanse of the sea? Scientists have studied the dynamics and optimization of search processes for decades, their interests ranging from military tasks such as locating enemy vessels or mines in the ocean and search strategies of animals for food to diffusion control of molecular processes in biological cells (1-4). Without prior knowledge about the location of the target, a searcher randomly explores the search space. However, as already argued by Shlesinger and Klafter (5), instead of performing a Brownian walk a better search strategy for sparse targets is that of a Lévy flight (LF): The agent moves randomly with a power-law distribution λðxÞ ' jxj −1−α of relocation lengths. Owing to their scale-free, fractal character, LFs combine local exploration with decorrelating, long-range excursions. These effect a reduced oversampling compared with Brownian search (i.e., the forager is less likely to return to previously visited sites). In the field of movement ecology (6), such random jump-like search processes are often referred to as "blind search" using saltatory motion, which is typical for predators hunting at spatial scales that exceed their sensory range (7-10). Such blind search, inter alia, was observed for fully aquatic marine vertebrate predators including plankton-feeding basking sharks (Cetorhinus maximus) (11), jellyfish predators, leatherback turtles (Dermochelys coriacea) (12), and southern elephant seals (13). This is the kind of search motion we investigate here.How commonly are LFs actually observed in nature? Apart from the flight of the albatross (14, 15), power-law relocation...