The official invasion that set the Russian-Ukraine conflict set in stone happened on February 24, 2022. You may ask, why even are they doing this?
1. What events led up to it?
There had already been some tension when Ukraine broke free from the Soviet Union in 1991. It hasn't always been this severe until most recently, in 2013, former president Viktor Yushchenko ignores EU and NATO agreements and instead tethers towards forming an economic tie with Moscow. The public did not see this well, forming protests; he was voted out of presidency, and Russia with rage, sends armed men to the small area of Crimea that's apart of Ukraine, and steals that territory. This was the major reason the conflict had worsened.
With a new leader, in 2019, Ukraine is determined to start fresh. Ties with the EU helps, and with Volodymyr Zelenskiy trying his hardest to join the NATO after Yushchenko abruptly stopped it.
However, Russia does not take this warmly. In fact, they start adding troops on the borders of Russia-Ukraine, saying their purpose is to run drills; obviously it's to seek intimidation and fear from the Ukraine citizens. They continue this for months, until they order NATO to bar Ukraine from joining, and to remove troops from especially Ukraine. Obviously, NATO does not take this well, and employs more defense, while the United States works to add more troops to surrounding NATO members such as Poland and Romania*.
Again, Putin did not like this. Declaring that Ukraine is a crucial part of Russia with history under Soviet Union, he declares attack with missiles and attacks on this dark day of February 24.
2. What does Putin want from this fiasco?
As mentioned earlier, Putin greatly wants Ukraine to be back where it was, under Soviet Union. He sees a ethnic history behind it, and wants to restore Russia to where it was before. Experts speculate that due to Ukraine having no help from NATO since they're not in it, it's a wonderful opportunity to get Russia to where it was before, since Putin is a very patriotic man for his country. Some people may even speculate that Putin is doing this to silence the freedom Ukrainians may have in their country. He believes that Russians may one day leave the country to resort to a more free and open option.
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Footnote:
*: NATO did not add troops to Ukraine since Ukraine is not a NATO country. While they had been trying to, especially since Zelenskiy's presidency, countries elected must have a unanimous vote, which Ukraine did not get.