The lyrics are definitely Australian - written by Andrew Barton 'Banjo' Patterson, one of Australia's greatest poets.
The tune is much more complicated. Originally, the song was played to a tune composed by Christina MacPherson in 1895. MacPherson said that she based it on a tune that she had heard before - Thou Bonnie Wood o' Criagielea which is a Scots tune. The thing is, there's actually not that much similarity at all between Criagielea and the original Waltzing Matilda score.
Regardless of that, though, the version that is most commonly played today is legally accepted to have been composed by Mary Cowan sometime before 1903 (the words were also slightly changed at that time, according to legend in order to advertise "Billy Tea").
This tune is also used for a British song, The Gay Fusileer but it's unclear which came first.
WALTZING MATILDA
Once a jolly swagman camped by a Billabong
Under the shade of a Coolabah tree
And he sang as he watched and waited till his billy boiled
"Who'll come a-waltzing Matilda with me?"
Down come a jumbuck to drink at the water hole
Up jumped a swagman and grabbed him in glee
And he sang as he stowed him away in his tucker bag
"You'll come a-waltzing Matilda with me'".
Up rode the Squatter a riding his thoroughbred
Up rode the Trooper - one, two, three
"Where's that jumbuck you've got in your tucker bag?",
"You'll come a-waltzing Matilda with me".
But the swagman he up and jumped in the water hole
Drowning himself by the Coolabah tree,
And his ghost may be heard as it sings in the Billabong,
"Who'll come a-waltzing Matilda with me?"
A.B. (Banjo) Paterson