No algorithm has been published that can factor all integers in polynomial time, that is, that can factor a $b$-bit number $n$ in time $O(b^k)$ for some constant $k$. Neither the existence nor non-existence of such algorithms has been proved, but it is generally suspected that they do not exist and hence that the problem is not in class P. The problem is clearly in class NP, but it is generally suspected that it is not NP-complete, though this has not been proven.
There are published algorithms that are faster than $O((1 + ε)^b)$ for all positive $ε$, that is, sub-exponential. As of 2021-03-12, the algorithm with best theoretical asymptotic running time is the general number field sieve (GNFS), first published in 1993, running on a $b$-bit number $n$ in time: [...] For current computers, GNFS is the best published algorithm for large $n$ (more than about 400 bits). For a quantum computer, however, Peter Shor discovered an algorithm in 1994 that solves it in polynomial time.
However, let's imagine for a moment that someday a classical algorithm for factoring all integers in polynomial time is discovered, would the interest in building quantum computer decrease?
Obviously, integer factorization is far from being the only application for quantum computers (I am thinking about Grover's algorithm, quantum mechanics simulations, etc), but Shor's algorithm has really turbocharged the interest in developing quantum computers.