fließend
fluent
Refers to the ability to speak a language effortlessly, correctly, and without hesitation.
Nach einem Jahr in Spanien sprach sie fließend Spanisch.
After a year in Spain, she spoke fluent Spanish.

fluent / flowing
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fluent
Refers to the ability to speak a language effortlessly, correctly, and without hesitation.
Nach einem Jahr in Spanien sprach sie fließend Spanisch.
After a year in Spain, she spoke fluent Spanish.

The word 'fließend' (fluent/flowing) is the present participle of the verb 'fließen' (to flow). This origin explains all its meanings: a language can 'fließen' like water (you speak effortlessly), liquids 'fließen', and transitions can also be 'fließend' (that is, seamless).
Der Redner beeindruckte durch seine fließende Ausdrucksweise.
The speaker impressed everyone with his fluent way of expressing himself.
When it comes to languages, 'fließend' (fluent) is almost always used as an adverb with verbs like 'sprechen' (to speak) or 'beherrschen' (to master). It describes how well someone speaks a language.
Sie spricht fließend drei Sprachen.
She speaks three languages fluently.
In the literal meaning, 'fließend' (flowing) is often combined with 'Wasser' (water) or 'Verkehr' (traffic). 'Fließendes Wasser' is a fixed term for running water from the tap.
Trotz der Hauptverkehrszeit war der Verkehr fließend.
Despite rush hour, the traffic was flowing.
To describe a seamless process, 'fließend' (smooth/flowing) is often combined with nouns such as 'Übergang' (transition) or 'Bewegung' (movement).
Die Choreografie zeichnete sich durch fließende Bewegungen aus.
The choreography was characterized by flowing movements.
ongoing / continual
'laufend' and 'fließend' can both describe movement without stopping, especially with water. However, 'laufend' is used more for water that comes from a pipe, a tap, or a spring. 'fließend' more generally describes a calm, steady movement, for example in a river, with water, or also with transitions. Learners may confuse the words because both have to do with movement. Important: with languages, you say 'fließend', not 'laufend'. 'laufend' also has other meanings, such as 'current' or 'continual'.

Maria spricht nach zwei Jahren in Deutschland fließend Deutsch.
After two years in Germany, Maria speaks German fluently.
In der alten Hütte gibt es kein laufendes Wasser.
There is no running water in the old cabin.