Should Children be Invited to the Wedding?

I love attending weddings where children of all ages are also invited. Sure, they can be mischievous, grumpy, messy…but, I love their little dressed up selves and how they stare wide-eyed at the beautiful bride. It’s fun to watch them play with their siblings and cousins, make new friends, and run all over the place.  They’re a lot of work; they’re a lot of fun!

I realize there are definitely circumstances where it would be unsafe to have younger children in attendance~ either at the wedding or at the reception. But, if weddings are a family gathering event, shouldn’t children be allowed to attend in most cases?
This very topic has caused many a division between family and friends when an invited guest has received a wedding invitation that excluded their children.  If the bride and groom decide that they want an adults only wedding and reception, it will be crystal clear by the invited names on the invitation envelope.  If that's the case, it is imperative that we as guests, be respectful and comply with their wishes~ no matter how we feel about it.
What is your opinion? Generally speaking, should children be invited to the wedding or not?