No, there are no official borders between continents.
There are several things to consider here in terms of what defines "continents":
- Cultural and ethnic matters, those purely on the "human" point of view
- Technical point of view, purely natural and geographical
- Politically, continental borders have, in some places, been arbitrarily decided without corresponding to either political or geographical boundaries (see below)
The concept of "continents" was invented in ancient Greece; in their viewpoint, the Aegean was the center of the world and was surrounded by three continents: Europe, Asia and Africa. However, they didn't have modern knowledge about tectonic plates, thus their borders were decided arbitrarily.
Asia, Europe and Africa are physically connected. It's erroneous to call them different continents. They're technically just one huge continent. However, long before humanity came into existence, the African plate was estimated to be separate from Eurasia, and the Indian subcontinent was separated from Asia, and still is on a separate tectonic plate.
By definition, maritime islands are not part of any continent. The only difference between a continent and an island is that a continent is arbitrarily larger than an island, but there is no clear border. It is debated whether Greenland and Australia are big islands or small continents. (Greenland appears huge on many world maps using a Mercator projection because it lies very close to the North Pole, but, in fact, it's not that huge.)
Many islands that are close to only one continent are frequently associated with the continent they're close to. For example, Great Britain is considered to be a part of "Europe" and Japan's islands are considered to be a part of "Asia," but this is due to purely cultural matters and in no case geographical matters. Obviously, this continental categorization of islands by proximity cannot be applied to islands that are either close to multiple continents or far from any continent.
Traditionally, the border between Europe and Asia is physically represented by the Bosphorus and the Dardanelles in present-day Turkey. However, the tradition of having a continental border there dates from ancient Greece.
In Russia, the European-Asian border is today considered to be the Ural mountains, and, to the south of that, the Ural river. Historically, the Don River, (much further west than the Ural River) was also used as a border. This move eastward seems to have been motivated by an interest in making Russia more of a "European" country. This new line of demarcation was proposed much more recently, by Vasily Tatishchev. Before his proposal there was no clear border; as such, this border is much newer than the physical boundary south of the Black Sea, which has existed for several thousands of years. The modern definition makes Kazakhstan yet another country crossing the continent border, as the Ural river flows through Kazakhstan, too, and a large portion of that country is west.
The Caucasus mountains serve as the border between the Europe-Asia border between the Black Sea and the Caspian Sea, but the mountain range is very wide, so there has never been any consensus as to where exactly the "border" passes. Before the first World War, Turkey and Russia bordered each other directly, so this was likely to be considered the border between Europe and Asia at that time.
The question about whether or not the newly independent Caucasian countries (Georgia, Armenia and Azerbaijan) are in Asia or Europe is subject to debate (perhaps even poisonous nationalist debate!) and based on opinion. The three countries are located south of the Caucasus, so they'd be "mostly Asian." However, Armenia and Georgia are traditionally Christian countries, so they tend to feel more European. On the other side, Azeris probably feel more Asian due to their shared heritage with Turkey and the Central Asian Turkic countries. All this is, however, debatable.
I'll add that Turkey was founded by Turks which invaded the region from Central Asia. Turkish is clearly an Asian language, and the people are of Asian culture. For example, they remove their shoes when entering a home, and talking loudly or disrespectfully is considered an absolutely awful practice, just like in Japan (whereas this behavior is common in all of Europe).
On the other hand, Russian is a Slavic language, which is clearly associated with Europe. The Asian part of Russia has been colonized by European Russians relatively recently. It is much less densely populated than the European part, where the vast majority of Russians live.
The border between Asia and Africa is today marked by the Red Sea and the Suez Canal. The canal was recently built, so before it was done, there was no clear border. Sometimes the Nile was used as a "continental border," and sometimes the eastern political border of Egypt, wherever it was during history, was used.
The border between Europe and Africa is physically delineated by the Strait of Gibraltar. Spain has two villages on the African side, so this also makes it a cross-continental country.
The OP didn't ask about it, but I'll still mention that the border between South America and North America has traditionally been the Panama Canal, in Panama. However, I'm fairly confident people from the USA and Canada consider all countries to the south of North America to be "South American" because of their Hispanic culture, even though they are living in North America. So once again, a cultural border does not match a geographical or historical border.