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Myself (UK citizen) and my husband (South African) currently live and work in the UAE.
He has never been to the UK and I would like to show him my native country for a short holiday of a few days, after which we will return to the UAE.

Because we are married, does he have to go through the usual, lengthy application process for a tourist visa, or is there an easier way we can apply? I’ve read threads on "family visa" but I'm struggling to get an answer to this particular situation.

The same goes for him for a Schengen visa. Is there a different, simpler process? He owns his own company in the UAE, but I am happy to act as his "sponsor" if it simplifies things.
Many thanks in advance, any advice would be much appreciated!

To enter the UK, he needs a standard visitor visa. Being married to a British citizen does not help, and may hurt, his application. The application should include evidence of his well established ties to the UAE.

Family visas are for those who want to immigrate, so they do not apply to your circumstances. They are quite expensive. An application from outside the UK costs £1523.

I do not wish to cause alarm by saying that being married to you may hurt his application. However, the entry clearance officer (ECO) examining his visa application must presume that he intends to immigrate to the UK and then see whether the evidence presented in the application is sufficient to rebut that presumption. For someone with a UK spouse, it can be more difficult to overcome the presumption, so evidence of your stable life in the UAE will also be helpful. In the current political climate, especially, ECOs can be unpredictable, so this needs to be said even if the chance of a refusal is low.

In his case, an established business owner coming to the UK for a brief visit with his wife who is also employed in the place where they live, the chance that he will be seen as an immigration risk seems low.

To enter any other EU or Schengen country, he can apply for a visa under the freedom of movement directive, 2004/38/EC, if he is traveling with you or joining you. This means that the application is free of charge and should be considered expeditiously, and that it may be refused only for reasons of public safety, public health, or public policy. Because of that, there are a few questions on the Schengen visa application that are marked as not applicable to those applying under the directive.

The free movement directive does not apply to your husband in the UK because national law generally takes precedence over European law in the relationship between a country and its own citizens. This means that non-EEA family of British citizens are generally subject to the immigration rules, as are non-EEA citizens generally, while non-EEA family of EEA citizens fall under the Immigration (EEA) Regulations, which implement the freedom of movement directive. There is an exception known as the "Surinder Singh" route, after the court case that created it, for British citizens who fall under the free movement directive because they have established themselves in another EU country. This seems unlikely to apply to you because you live in the UAE. But if you have lived together in another EU country in the past, you should look into it more closely.