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We will discuss the generalized Fermat equations $Ax^2 + By^{2n} = 4z^3$, assuming (for simplicity) that the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-AB})$ is one. The methods use techniques coming from Galois representations and modular forms; for small $n$'s one needs Chabauty type methods. Our results, conjectures (and methods) extend those given by Bruin, Chen et al. in the case $x^2 + y^{2n} = z^3$. This is a joint work  with K. Chałupka and G. Soydan.