Yes, it is forbidden by the Torah in two ways. First, there is a prohibition against cohabiting while the woman has the status of niddah. Single Jewish women have that status until they immerse in a mikveh prior to marriage. Second, the Rambam understands the Torah prohibition of kedeshah (prostitution) to prohibit any sexual relations outside of the context of marriage.

There is no room for a woman to acquire the status of pilegesh nowadays.

Being physically intimate with a man, even if the couple does not have relations, is also a violation of the Torah. The Torah prohibits not only relations with a woman in niddah, but any act that approaches it. Approach is understood to cover a broad array of physically affectionate behaviors.

As challenging as it may be to refrain from physical relations prior to marriage, restraint is demanded by halacha. Halacha has a positive view of physical intimacy and its potential for joy and even holiness. However, this potential is reached within the halachic framework and commitment of marriage.

Please do not hesitate to write back with any additional questions.