It was the question first raised when Newton for the first time postulated the theory of gravity. Then many contemporary scientists refused the Newtonian theory arguing that if gravity is real then all universe must collapse in one point. To defend his theory newton for the first time purposes that universe is infinite and in infinite universe there is no any point to fall all body. Then after some time this theory was revised by arguing that gravity is a force that work as attractive in short range and repulsive in long range. But this theory was ignored. It took too much time to understand the real nature of gravity. It was Albert Einstein who postulated the general theory of gravity that finally solved the question about gravity. At first Einstein  also confused about this. and he add cosmological constant in  his theory. But after Hubble found universe is expanding he remove the cosmological constant. this really solve the problem of gravity but still there is question about universe. It is finite or infinite? Some argues it as finite and some as infinite. what is the reality? Nobody knows.
But i think the universe is spatially infinite. I will take the help of uncertainty principle. which is as follows
The uncertainty principle is the concept that precise, simultaneous measurement of some complementary variables -- such as the position and momentum of a subatomic particle -- is impossible. Contrary to the principles of classical physics, the simultaneous measurement of such variables is inescapably flawed; the more precisely one is measured, the more flawed the measurement of the other will be."
By this it is clear that both position and moment of universe can not zero. this mean if universe is finite then it implies that both position and momentum is zero outside the universe. So it will violate the uncertainty principle. so universe must be spatially infinite.