I'm a dual citizen of Britain and Australia (we live and always have lived in Australia). My husband and I want to travel in Schengen countries without having to worry about 90 day limit or hopping to countries outside the Schengen to comply with 90 day stay in 180 days. As he's married to me, a British citizen, I wondered if there any way we can avoid applying for visas?
While the UK remains in the EU, you can travel together under the freedom of movement directive, 2004/38/EC. This means that the 90/180 rule does not apply to your husband because is not considered a "third-country national" under the Schengen Borders Code. Article 2(6):
‘third-country national’ means any person who is not a Union citizen within the meaning of Article 20(1) TFEU and who is not covered by point 5 of this Article;
‘persons enjoying the right of free movement under Union law’ means:
(a) Union citizens within the meaning of Article 20(1) TFEU, and third-country nationals who are members of the family of a Union citizen exercising his or her right to free movement to whom Directive 2004/38/EC of the European Parliament and of the Council ( 1 ) applies;
Directive 2004/38/EC provides that if you stay in any one country for over three months, that country can require you to meet the criteria specified in Article 7. They can also require you to register, under Article 8, and they can require your husband to apply for a residence card, under Article 9.
If you do not reach the three-month threshold in any country, your stay there falls instead under Article 6, which is fairly short and to the point, as far as legislation goes. In its entirety:
Right of residence for up to three months
Union citizens shall have the right of residence on the territory of another Member State for a period of up to three months without any conditions or any formalities other than the requirement to hold a valid identity card or passport.
The provisions of paragraph 1 shall also apply to family members in possession of a valid passport who are not nationals of a Member State, accompanying or joining the Union citizen.
I should warn you, however, that at least some governments have put forth a different interpretation of this legislation, which has been covered in a question and answer here that I can't find right now. I'm also involved in an e-mail exchange with Your Europe Advice, which was the source in that question and answer of some incorrect advice. When I find the question again, I'll update this answer.
The practical implication of that varying interpretation is that some countries might try to fine your husband improperly. I can say from my parents' experience that France is not one of those countries. If they do fine him, you should be able to challenge that successfully in court, but that is a burden you probably do not want to face. The chance of a fine actually being imposed is probably pretty low, however.
After the UK leaves the EU, and after the end of the planned transition period, you will both be subject to the 90/180 rule. The situation for UK visitors to the EU during the transition is unclear, but my guess is that they will also be subject to the 90/180 rule during that time.