We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial f= y n+∑ i= 0 n− 1 a i (x) y i∈ F q [x][y] with iid coefficients a i taking values in the set {a (x)∈ F q [x]: deg⁡ a≤ d} with uniform probability, is irreducible with probability tending to 1− 1 q d as n→∞, where d and q are fixed. We also prove that with the same probability, the Galois group of this random polynomial contains the alternating group A n. Moreover, we prove that if we assume a version of the polynomial Chowla conjecture over F q [x], then the Galois group of this polynomial is actually equal to the symmetric group S n with probability tending to 1− 1 q d. We also study the other possible Galois groups occurring with positive limit probability. Finally, we study the same problems with n fixed and d→∞.