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This paper deals with different models of random walks with a reinforcedmemory of preferential attachment type. We consider extensions of the ElephantRandom Walk introduced by Sch\"utz and Trimper [2004] with a strongerreinforcement mechanism, where, roughly speaking, a step from the past isremembered proportional to some weight and then repeated with probability $p$.With probability $1-p$, the random walk performs a step independent of thepast. The weight of the remembered step is increased by an additive factor$b\geq 0$, making it likelier to repeat the step again in the future. Acombination of techniques from the theory of urns, branching processes and$\alpha$-stable processes enables us to discuss the limit behavior ofreinforced versions of both the Elephant Random Walk and its $\alpha$-stablecounterpart, the so-called Shark Random Swim introduced by Businger [2018]. Weestablish phase transitions, separating subcritical from supercritical regimes.

On a class of random walks with reinforced memory