The KPSS tests the null hypothesis of (short memory) stationarity. You can even apply the KPSS after you take the rank-order of your time series (see my paper on the Journal of Econometrics 172(1), http://www.sciencedirect.com/science/article/pii/S0304407612002151) to have a more robust test.

The sample ACF is not necessary, even though short memory stationary time series have ACF that converge to zero at exponential rate.

Hiba another unit root test is the Phillips-Perron test. The ACF is often used to model an I(0) series after, and not before, investigating the stationarity status of the series.  It is never used to ascertain the unit-root status of the series. 

Thank you Matteo and Ette for your answers.

Testing for stationary in time series, you used the tools for it such as ADF. KPSS,  Ng-Perron, Philips-Perron e.t.c But the ACF and PACF are used for models selection

ADF statistically test the stationarity-in-mean for data series, and acf and pacf confirm the result of the stationary graphically. The results from adf and the correlograms must be consistent for the data series.

In addition to what others mentioned we have also the following unit root tests:

1. Zivot-Andrews Unit Root test allowing for a single break in intercept and/or trend

2. Clemente, Montanes, Reyes unit root tests with two structural breaks

3. Clemente, Montanes, Reyes unit root tests with one structural break

Thank you Samson, Siti and Shimelis for your valuable responses. However, I am not a statistician, so I am a little confused about the difference between these various unit root tests and what each one implies and how does that contribute to determining whether or not the data series is stationary? I am also confused about the definition of stationarity: that we need to check the ensemble mean!  Will someone explain please. 

Thank you.

ACF and PACF are mainly for model identification after stationarity test and deferencing of nonstationary series (data).

Hiba Ali ACF and PACF are mainly for model identification after stationarity test and deferencing of nonstationary series (data).