Positive to negative or negative to positive?
Yes.
Before a lot was known about electrons it was thought that current flows out of the positive terminal of the battery, through the external circuit components, and back into the negative terminal of the battery. We now know that electrons are actually flowing in the opposite direction but in general we use the original assumption when we talk about current flow.
As for my 'Yes' answer above ... Inside the battery current is flowing from the negative terminal to the positive terminal so there room for confusion with your question as posed.
Don
What's the convention on current flow? Positive to negative
Positive to negative flow is called conventional current and is what every one in electronics uses.
The other direction is know as electron flow and is used only when you want to consider the actual electrons.
Yes that is what it means.
As you rightly say, it matters not a jot as to which direction you consider it flows since the signs (direction) will balance. ie the sum of the currents at any point equals zero. However you must stick to a single convention and that generally is from the positive terminal of a current supply source. The term electron flow is not the same as the term current flow. Us oldies even had to consider "hole flow" when dealing with transistor theory.
What you need to consider is whether you are considering a current source or a current sink.
A battery is a current source so you can/should consider the current comes OUT of the positive terminal. However a device such as the arduino which has + and - power supply connections is a current sink. In this case the curent flows INTO the positive terminal. BUT, where the arduino has positive and negative output terminals, the current flows OUT of the + terminal since these terminals are considered as current sources.
All very confusing until you get your head around sources and sinks.
@jackrae I would disagree with this, sourcing and sinking are terms that refer to if current is controlled next to the ground or next to the supply. Electron flow is never considers in solid state electronics. The legacy of electron flow is however found in the naming of the transistors emitter and collector and the FETs source and drain, but these are just names.
Hi Mike
It's your choice - but it works for me
For every answer there is at least one which is equal and opposite (with apologies to Isaac Newton)
Jack
For every answer there is at least one which is equal and opposite (with apologies to Isaac Newton)
And... Just because two people argue doesn't mean that one of them must be right. ]
In the end it depends if the current is made of positive or negative charge. All of this and more is very well described in Wikipedia: Electric current - Wikipedia
No it always flows in the same direction. And by it I mean current.
This is all totally irrelevant to the OP question.
Oops, he was asking for the convention --> you are definitely right.
It is really, mostly, semantics, BUT,,,
Current does not flow, current IS flow.
Electron current/flow is the observation of the movement of electrons (with negative charge, negative opposes negative, attracts positive) from negative source to positive sink.
Conventional current is the observed flow (or current) of Holes left in the atomic shells of the conductor material by the bumping off of electrons. The holes represent a positive charge, because of the absense of the electrons, and 'flow' from positive to negative ( positive rejects positive and attracts negative).
Wow, 12 years of college and tech schools did do some good,,,, my brain hurts.
Grumpy_Mike:
What's the convention on current flow? Positive to negative
Positive to negative flow is called conventional current and is what every one in electronics uses.
The other direction is know as electron flow and is used only when you want to consider the actual electrons.
I was schooled in the concept that conventional current flow (positive to negative) was defined (incorrectly) by the early pioneers of electricity as they knew that current flowed in one direction only in a DC circuit, but guessed wrong on the actual direction because they knew nothing about atoms and their structure. But the legacy continued to this day in classic electrical engineering. Later as the atom was better understood it was clear that what moves in current flow is electrons from atom to atom via the valence ring and that they move from negative charge to positive.
So colleges continued to teach classic conventional current flow and even 'invented' the 'hole' flow analogy to make their flawed concept work. In my thinking a electron is a physical thing, it has mass and can actually move either by atom to atom or as a stream emitted from a hot emitter. A 'hole' on the other hand is not a physical thing, has no mass and thinking it can 'flow' from atom to atom is rubbish to me. 
In U.S. Military electronics training we were taught to use electron flow and that it was negative to positive and to not be confused about all the backward pointing arrows used in semiconductor symbols.
It's always a nice subject to argue about, but like masturbation the end result always leaves one....spent but not completely satisfied.
Lefty
So your first paragraph essentially reinforces what I said back in reply #1....
Don
and even 'invented' the 'hole' flow analogy to make their flawed concept work
No that was not the reason for hole flow. It came about because of doped semiconductors where flow in P type semiconductors is actually easier to describe in terms of hole flow, than electron flow.
Now get this holes even have mass, in certain parts of the conduction band. This is the inverse of the electrons that have negative mass at the same point. A negative mass is one that is repelled by a gravitational field instead of being attracted to it. Holes also carry positive charge, so holes are just as real as electrons.
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floresta:
So your first paragraph essentially reinforces what I said back in reply #1....
Don
Great minds thinking alike?
Grumpy_Mike:
and even 'invented' the 'hole' flow analogy to make their flawed concept work
No that was not the reason for hole flow. It came about because of doped semiconductors where flow in P type semiconductors is actually easier to describe in terms of hole flow, than electron flow.
Now get this holes even have mass, in certain parts of the conduction band. This is the inverse of the electrons that have negative mass at the same point. A negative mass is one that is repelled by a gravitational field instead of being attracted to it. Holes also carry positive charge, so holes are just as real as electrons.
That is mostly rubbish talk in my rather worthless opinion. Where is the gravitational field in space where electronics work just dandy thank you. What is the mass quantity of a hole? Gravity is still a subject that science still has but a weak explanation for.
Lefty
It might sound rubbish but that's quantum Physics for you.
Gravity is all around everywhere actually. 
I fail to grasp the gravity of all this mad rubbish.....
oes this mean that super massive black holes actually have no mass, or do they only conduct in dark wire systems???
My brain hurts even more now......
General relativity is a weak explanation for gravity and the absence of an electron in an orbital and the forces involved are some kind of rubbish?
Oh wait, did I check my horoscope? Did I like, channel from the home planet with the right crystal? Did I get the real answers while meeting the groovy minds up on the astral plane? Like, are the vibes right or not? Ohhhh, wow! Yeah, science is such a downer, man. All that math and stuff just sucks! Like you know, the Tea Party is really onto something, we should all go back to like before all that stuff and get our Aristotle-heads on.
GoForSmoke:
General relativity is a weak explanation for gravity and the absence of an electron in an orbital and the forces involved are some kind of rubbish?
Oh wait, did I check my horoscope? Did I like, channel from the home planet with the right crystal? Did I get the real answers while meeting the groovy minds up on the astral plane? Like, are the vibes right or not? Ohhhh, wow! Yeah, science is such a downer, man. All that math and stuff just sucks! Like you know, the Tea Party is really onto something, we should all go back to like before all that stuff and get our Aristotle-heads on.
Note that there are several high end audio cable manufactures that sell (to the golden ear audio customers) their line level audio interconnect cables with arrows marked on them showing the proper direction of signal flow for best audio results. They too give a complex explantion for why an AC audio signal needs to use a 'directional' cable. Snake oil still sells in some markets.
I have no qualm with the subject of semiconductor physics, they must know what they are talking about as there is no shortage of very cool devices that keep becoming avalible. I just question the concept that if I short out the terminals of a battery with a wire that two things then flow, electrons and 'holes'.