The decades-old feud between them is exacerbated by religious differences. They each follow one of the two main sects in Islam – Iran is largely Shia Muslim, while Saudi Arabia sees itself as the leading Sunni Muslim power.
Historically Saudi Arabia, a monarchy and home to the birthplace of Islam, saw itself as the leader of the Muslim world. However this was challenged in 1979 by the Islamic revolution in Iran which created a new type of state in the region – a kind of theocracy – that had an explicit goal of exporting this model beyond its own borders.
The 2003 US-led invasion of Iraq overthrew Saddam Hussein, a Sunni Arab who had been a major Iranian adversary. This removed a crucial military counter-weight to Iranian influence in Iraq, which has been rising since then.
Fast-forward to 2011 and uprisings across the Arab world caused political instability throughout the region. Iran and Saudi Arabia exploited these upheavals to expand their influence, notably in Syria, Bahrain and Yemen, further heightening mutual suspicions.
Iran’s critics say it is intent on establishing itself or its proxies across the region, and achieve control of a land corridor stretching from Iran to the Mediterranean
How have things suddenly got worse?
The strategic rivalry is heating up because Iran is in many ways winning the regional struggle.
Saudi Arabia is trying desperately to contain rising Iranian influence and the militaristic adventurism of the kingdom’s young and impulsive Crown Prince Mohammed bin Salman – the country’s de facto ruler – is exacerbating regional tensions.
He is waging a war against rebels in Saudi Arabia’s southern neighbour, Yemen, in part to stem perceived Iranian influence there, but after nearly three years this is proving a costly gamble.
Meanwhile in Lebanon, many observers believe the Saudis put pressure on the prime minister to resign in order to destabilise a country where Iran’s ally, Shia militia group Hezbollah, leads a politically powerful bloc and controls a huge, heavily armed fighting force.
There are also external forces at play. Saudi Arabia has been emboldened by support from the Trump administration while Israel, which sees Iran as a mortal threat, is in a sense “backing” the Saudi effort to contain Iran.
The Jewish state is fearful of the encroachment of pro-Iranian fighters in Syria ever closer to its border.
Israel and Saudi Arabia were the two countries most resolutely opposed to the 2015 international agreement limiting Iran’s nuclear programme, insisting that it did not go far enough to roll back any chance of Iran obtaining the bomb.
Who are their regional allies?
Broadly speaking the strategic map of the Middle East reflects the Shia-Sunni divide.
In the Iranian camp is Syria’s government, which has been strongly backed by Iran, and where pro-Iranian Shia militia groups, including the Lebanon-based Hezbollah, have played a prominent role in fighting predominantly Sunni rebel groups.
The Shia-dominated Iraqi government is also a close ally of Iran, though paradoxically it also retains a close relationship with Washington on whom it has depended for help in the struggle against so-called Islamic State.
How is the Saudi-Iranian rivalry being played out?
This is in many ways a regional equivalent of the Cold War, which pitted the US against the Soviet Union in a tense military standoff for many years.
Lebanon risks being tipped into Syria-like chaos but few analysts see Saudi interests prevailing there.
Are we heading towards a direct war between Saudi Arabia and Iran?
So far Tehran and Riyadh have fought via proxies. Neither is really geared up for a direct war with the other but one successful rocket attack on the Saudi capital from Yemen could upset the apple cart.
One obvious area where they could come into direct conflict is in the waters of the Gulf, where they face each other across a maritime border.
But here too fighting could risk a much broader conflict. For the US and other Western powers, freedom of navigation in the Gulf is essential and any conflict that sought to block the waterway – vital for international shipping and oil transportation – could easily draw in US naval and air forces.
For a long time the US and its allies have seen Iran as a destabilising force in the Middle East. The Saudi leadership increasingly sees Iran as an existential threat and the crown prince seems willing to take whatever action he sees necessary, wherever he deems it necessary, to confront Tehran’s rising influence.
The danger is that Saudi Arabia’s new activism is fast making it a further source of volatility in the region.