
beherrschen
to master
To be very good at a skill, body of knowledge, or language and use it confidently.
Sie beherrscht drei Fremdsprachen fließend.
She masters three foreign languages fluently.
to master / control / dominate
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to master
To be very good at a skill, body of knowledge, or language and use it confidently.
Sie beherrscht drei Fremdsprachen fließend.
She masters three foreign languages fluently.

to control oneself
To keep one's feelings or impulses under control and not show them; to keep one's composure.
Obwohl er wütend war, konnte er sich gerade noch beherrschen.
Although he was angry, he just managed to control himself.

to rule
To exercise power over and control a territory, a people, or a group.
Der König beherrschte sein Reich mit strenger Hand.
The king ruled his realm with a firm hand.

to dominate
To be the dominant or defining element of something, for example in a landscape or a topic.
Die riesige Burg beherrscht die ganze Landschaft.
The huge castle dominates the whole landscape.
'Beherrschen' derives from 'der Herr' (the lord/master) and 'herrschen' (to rule). The prefix 'be-' makes the verb transitive, so it always needs an accusative object. The basic idea is to have full control over something.
Ein guter Reiter beherrscht sein Pferd.
A good rider controls his horse.
The meaning changes completely depending on whether the verb is reflexive or not. 'Sich beherrschen' means to control one's own feelings. 'Etwas beherrschen' means to master a skill or rule/control an area.
Er war so wütend, dass er sich kaum beherrschen konnte.
He was so angry that he could hardly control himself.
In the meaning of 'etwas können' (to be able to do something), 'beherrschen' is often combined with skills and knowledge: 'eine Sprache beherrschen' (to master a language), 'ein Instrument beherrschen' (to master an instrument), 'eine Methode beherrschen' (to master a method), or 'ein Handwerk beherrschen' (to master a craft).
Mein Großvater beherrscht das Handwerk des Schreinerns.
My grandfather has mastered the craft of carpentry.
'Sich beherrschen' is often used in situations where someone suppresses strong emotions. Common phrases include 'sich nicht mehr beherrschen können' (to no longer be able to control oneself) or 'sich nur mit Mühe beherrschen' (to control oneself only with difficulty).
Als sie die Beleidigung hörte, konnte sie sich nicht mehr beherrschen.
When she heard the insult, she could no longer control herself.
Whether 'beherrschen' means 'regieren' (to rule) or 'dominieren' (to dominate) depends on the object. If it refers to people or a country, it means 'regieren' (to rule). If it refers to a landscape, a topic, or a discussion, it has the figurative meaning 'dominieren' (to dominate).
Die Diskussion wurde von einem einzigen Thema beherrscht.
The discussion was dominated by a single topic.
to rule / prevail
Both herrschen über and beherrschen can express having control over one's own feelings. Beherrschen is the usual and more direct form for this, while herrschen über sounds more marked and describes control as a kind of power. In normal conversation, beherrschen is usually the better choice; herrschen über is also possible, but less neutral.

Mit großer Disziplin beherrschte er seine Gefühle und blieb ruhig.
With great discipline, he controlled his feelings and remained calm.
Mit großer Disziplin herrschte er über seine Gefühle und blieb ruhig.
With great discipline, he kept his feelings under control and remained calm.