Arithmetic derivative is the function in number theory, sending each prime into 1 and satisfying the Leibniz rule $D(ab) = D(a)b + aD(b)$ for any $a,b$. The corresponding dynamical system: $n \rightarrow D(n)$ has two obvious attractors: $0$ and $\infty$. One of the major conjectures about arithmetic derivative is that the corresponding directed graph, whose vertices correspond to nonnegative integers and whose arcs connect a number $n$ with its derivative $D(n)$ contains no cycles except the loops in fixed points $p^{p}$, where $p$ is any prime. The purpose of the talk is i) to give a brief review of what is known or conjectured about arithmetic derivative, ii) to define some related new concepts and iii) to present some new results.