First things first: genealogical relationship
The Ukrainian language is one of the Slavic languages of the Indo-European family of languages which includes most of the languages of Europe and a number of Indo-Iranian languages. All these have a theorized common ancestor -- the Proto-Indo-European (PIE) language that linguists have been reconstructing since the early 19th century based on the history of documented languages of the family.
Languages of Europe

The origin of Indo-Europeans

The Pontic–Caspian steppe north of the Black Sea, which includes the south of Ukraine, is widely believed to have been the homeland for nomadic proto-Indo-Europeans, who domesticated horses, which allowed them to migrate across the continent in wagons. The area is related to Yamna archaeological culture (3300–2600 BC) associated with the burial mounds known as kurgans.
Other theories suggest that Proto-Indo-Europeans originated in Anatolia (modern Türkiye), Caucasus, or India. Whatever is the place of origin, it is still impossible to make any substantiated suggestion as to where the proto-Indo-Europeans could have arrived from and what languages they could have spoken before PIE emerged.
Every language, unless it's isolated geographically or dead, interacts with neighboring languages throughout its entire history, exchanging words, grammar and pronunciation features.

Ukrainianisms in other languages
Together with the maidan, borshch, Cossack, and pysanka mentioned above, most of the Ukrainian words that made their way to other languages are related to Ukrainian culture, history, ethnic, or geographic terms.
The term steppe came to English via German from Russian around the 17th century. The original etymology of the word steppe is unknown, however, the Slavic dialects that later formed Russian were located mostly in the north far from steppe areas so Russian could have borrowed the word from older Ukrainian dialects. Another term of Ukrainian origin that came to other languages via Russian is kurgan "tumulus" (kurhan in Ukrainian pronunciation).
With borshch, culinary terms also include horilka 'Ukrainian vodka,' kasha 'porridge,' holubtsi 'cabbage rolls,' varenyky 'dumplings,' syrnyky 'quark cheese pancakes.'
With Cossack, historic terms include hetman 'highest military officer,' Sich 'administrative and military center of the Zaporozhian Cossacks.' The words boyar 'nobleman in Eastern Europe in the 10th-17th centuries' (Ukrainian boyaryn) and knyaz 'prince, duke, count' date back to Kyivan Rus times. Knyaz, by the way, has its roots in proto-Germanic or in Gothic, but let's talk about Goths later.

Influences on Ukrainian in modern times and in recent centuries
As English has been the world's lingua franca or the de-facto language of international communication since the 20th century, most of the loanwords nowadays come to Ukrainian from or via English. Examples of such words are menedzher 'manager,' biznes 'business,' kompyuter 'computer,' samit 'summit (meeting),' pled 'plaid', and so on and so forth.


Between Kyivan Rus and annexation by Russian empire (13-17th c.)
In this period Ukrainian fully formed as a spoken language, developing most of its distinctive features to re-emerge as a literary language sometime later.
An interesting Latin loanword of the time period is a colloquial term for vodka, okovyta, which is a distorted Latin phrase aqua vitae 'water of life.' Cognates of the phrase are also present in Polish and Belarusian.

Kyivan Rus

Pre-Kyivan Rus times and antiquity
Among the earliest borrowings from Turkic languages were losha 'foal,' sahaydak 'quiver,' karyi 'dark brown (eyes)' which came due to the contacts with Polovtsy and Pechenegs and other Turkic tribes that lived in modern Ukrainian steppes at the times. The earliest borrowings from Greek came via direct Slavic contacts with colonists in Greek cities on the Black Sea coast and in Crimea, as well as via other languages. Several examples of the early borrowings from Greek are kyt 'whale,' korabel 'ship,' myata 'mint,' ohirok 'cucumber,' troyanda 'rose flower,' myska 'bowl,' parus 'sale,' lyman 'estuary,' vyshnia 'cherry.' Due to rare contacts with Romans, eastern Slavs didn't have a lot of early borrowings from Latin, rare Latinisms are koliada 'carol' from Latin kalendae 'first day of the month,' and later kesar 'caesar,' and fortuna 'luck.' Meanwhile, the Medieval Latin word sclāvus 'slave' derived from Byzantine Greek Σκλάβος 'Slav' (after Sklavins, the name of one of the Slavic tribes). The late Latin word emerged as Slavs were often targeted in the early Middle Ages for enslaving by conquering peoples.'Not that modern parody, the real Goths'
The Proto-Slavic common language started to collapse around the 4th century. So most of the developments before this date, including borrowings, are shared by most of the Slavic languages.
Going deeper: Celtic and Iranian borrowings
In their prime as of 275 BC, Celtic tribes spanned across most of Europe from the Atlantic up to the west of modern Ukraine, while the Western Scythians, generally believed to have been of Iranian origin, dominated the Pontic steppe at the time. Unfortunately, both didn't leave behind any written documents in the territory of modern Ukraine, however, traces of their languages retained in the Ukrainian toponymics - names of the places. For example, the name of the small river Radorobel in Zhytomyr Oblast is of Celtic descent meaning 'one having rapids' while Ukraine's largest river - Dnipro/Dnieper bears the Scythian name known via Greek sources as Δανάπρις which means 'deep water.' In total, some 200 geographic names in Ukraine are believed to be of Celtic origin. Some linguists argue that the Ukrainian word motyka 'mattock' is borrowed from Celtic, maybe via Germanic languages. Meanwhile, the Old Slavic word vladyka 'lord, sovereign, archbishop' now mostly used in its religious meaning has a Celtic parallel, particularly Welsh gwledych 'kingdom, government, reign, ruler'. Welsh gwlad means 'country, land, region,' meanwhile the Slavic word is generally believed to be derived from vlada '(political) power' so the words seem to be native in both language groups, nevertheless, the Slavic term could have emerged by adapting the foreign term based on the native lexical resources. Researcher Václav Blažek draws parallels between Old Irish deity Dagdae (Celtic *dago-dēuo- ‘good god’) and Slavic god Dazhboh (from *Dadjьbogъ "God give").
As any modern language, Ukrainian has a long history of transformations in itself and of various interactions with other languages, which includes borrowings. The further back we look into the history of a language, the less we can be sure as we face more and more of the unknown.Probably the most interesting unknown in the Ukrainian ancient history is who were the representatives of the Trypillya culture, a Neolithic archaeological culture of the 5th- millennium BC that covered substantial parts of modern western Ukraine, northeastern Romania, and Moldova. These pre-historic farmers were building the largest settlements in Neolithic Europe and disappeared shortly after then-nomadic Indo-Europeans emerged. But this is a story for another time.
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