The number of square-free integers in N consecutive values of any polynomial f is conjectured to be c_f*N, where the constant c_f depends only on the polynomial f. This has been proven for degree less or equal to 3, but only conditional results are known for degree greater than 3. In 2013 Igor Shparlinski proved that this conjecture holds on average over all polynomials of a fixed naive height. In my talk I will describe a new general method first developed by Timothy Browning, Efthymios Sofos and Joni Teräväinen, and derive new results with less averaging from equidistribution results of arithmetic progressions in short intervals.