Summary: In this paper we define the fractional Cox-Ingersoll-Ross process as \(X_t := Y_t^2 \mathbf{1}_{\{ t < \inf \{s > 0:Y_s = 0\}\}}\), where the process \(Y = \{Y_t,t \geq 0\}\) satisfies the SDE of the form \(dY_t=\frac{1}{2}(\frac{k}{Y_t}-aY_t)dt+\frac{\sigma}{2}dB_t^H\), \(\{B^H_t, t \geq 0\}\) is a fractional Brownian motion with an arbitrary Hurst parameter \(H\in(0,1)\). We prove that \(X_t\) satisfies the stochastic differential equation of the form \(dX_t=(k-aX_t)dt+\sigma\sqrt{X_t}\circ dB_t^H\), where the integral with respect to fractional Brownian motion is considered as the pathwise Stratonovich integral. We also show that for \(k>0\), \(H>1/2\) the process is strictly positive and never hits zero, so that actually \(X_t=Y_t^2\). Finally, we prove that in the case of \(H<1/2\) the probability of not hitting zero on any fixed finite interval by the fractional Cox-Ingersoll-Ross process tends to 1 as \(k\rightarrow\infty\).