Is there really current in AC current?
In an AC system the mean voltage and the mean current are both zero but just because the mean of a quantity is zero doesn't mean that the quantity doesn't exist. 

As an analogy consider two hydralic cylinders connected together by a pipe. I push one cylinder in and pull it out again. There was no net flow in the pipe but there was clearly flow when I was pushing the cylinder in and also flow in the opposite direction when I was pulling it out. If I place a load on one cylinder and move the other up and down i'm clearly using the flow of water to transfer power even though there is no net water flow.

Getting back to electricty consider a resistor connected to a sinewave AC power supply. For simplicity lets assume that our voltage source has a peak voltage of 1 unit and our resistor has a resistance of one unit. We can write equations for the instantaneous voltage, instantaneous current and instantaneous power.

V=sin(2πft)
I=V/R=sin(2πft)
P=IV=(sin(2πft))&lt;sup&gt;2&lt;/sup&gt;


If we calculate the mean values of those functions we quickly see that V and I have a mean of 0 but P doesn't, it has a mean of 0.5. 

So if our mean voltage and mean current are zero how do we meaningfully measure voltage and current? We use something called the "root mean square" value. That is we take the voltage or current waveform, square it, take the mean and then take the square root. Voltages and currents in AC power system are nearly always measured and discussed in RMS values.

We can use these RMS voltage and current values in the equations P=I2R and P=V2/R equations to calculate average power delivered to a resistive load.

It's tempting to also think we can apply RMS values to the P=IV equation but it is important to remember that this only applies if the voltage and current waveforms are proportional. 

Further reading: https://en.wikipedia.org/wiki/Root_mean_square#Average_electrical_power