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Let $(M,J)$ be a $2n$-dimensional almost complex manifold and let $x\in M$.We define the notion of almost complex blow-up of $(M,J)$ at $x$. We prove theexistence of almost complex blow-ups at $x$ under suitable assumptions on thealmost complex structure $J$ and we provide explicit examples of such aconstruction. We note that almost complex blow-ups are unique. When $(M,J)$ isa $4$-dimensional almost complex manifold, we give an obstruction on $J$ to theexistence of almost complex blow-ups at a point and prove that the almostcomplex blow-up at a point of a compact almost K\"ahler manifold is almostK\"ahler.

Almost complex blow-ups and positive closed $(1,1)$-forms on $4$-dimensional almost complex manifolds