Number and Recent Evolution
In recent years, the LGBTQ+ community has continued to fight discrimination whilst campaigning for an improvement in equality and civil rights. In the UK, more recently we have seen the recognition of both civil partnerships, same-sex marriage and even of equality for same-sex child adoption.
The natural consequence of the countries’ acknowledgment of same-sex marriage and civil partnerships is that more and more LGBTQ+ people are looking for ways to grow their family via adoption or to have their own biological children.
The Marriages (Same Sex Couples) Act 2013 is an important piece of legislation for same-sex couples in England & Wales as it allows them to marry in a civil ceremony (i.e. in a register office) or on religious premises (provided that the religious organisation agrees). Additionally, this Act enables civil partners to convert their partnership to a marriage, if they wish.
According to The Office for National Statistics (ONS), more than 35,000 same sex couples have legally married since this piece of legislation came into force in March 2016. Remarkably the same source confirms that as many as 68% of LGBTQ+ people remain unmarried in contrast with the national statistic of only 16%.
This difference is perhaps due to the fact that the law changed only recently and current moral perceptions have yet to comprehensively shift. However, experts are predicting this gap will get smaller in the next few decades.
In respect to the welfare of offspring in LGBTQ+ families, it is important to confirm that the American Society for Reproductive Medicine (which is a multidisciplinary organization dedicated to the advancement of the science and practice of reproductive medicine) found no persuasive evidence that children are harmed or disadvantaged solely by being raised by single parents, unmarried parents, or LGBTQ+ parents. It did not support restricting access to assisted reproductive technologies on the basis of a prospective parent's marital status or sexual orientation.
Here a few recommendation links that might help you to find your best programme:
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Men having Babies is an USA Charity to support Gay men to became parents by surrogacy and developed an extensive and very useful data collection about agencies and IVF Clinics: http://www.menhavingbabies.org/surrogacy-directory/
Facebook groups are nowadays one the most used channels to find surrogate. It is high risk and less protection but in some countries it is sadly the only option.
Russian courts have confirmed the right of single men to undertake surrogacy and many openly gay men have exercised this right. Nevertheless, Russia began to receive worldwide attention for their legal treatment of LGBT citizens, which has been referred to as one of the worst human rights violations in the post-Soviet era.
Rest of the World
Co-parenting refers to a parenting situation where adults share the duties of parenting a child. Its original meaning was mostly related to united families, that a child has to continue to maintain a strong relationship with both parents, even if separated, has become more and more a recognized right. Thus, the concept of co-parenting was extended to separated and divorced families as well. Such arrangements are also now common among gay and polyamorous parents.
Here some recommended sites:
 The Ethics Committee of the American Society for Reproductive Medicine (2009). "Access to fertility treatment by gays, lesbians, and unmarried persons". Fertil. Steril. 92 (4): 1190–3. PMID 19732884. doi:10.1016/j.fertnstert.2009.07.977