〜ていく・〜てくる
Change over time; direction of an actionmachine translation
FormationVerb て-form + いく/くる
〜てくる expresses a change from the past up to the present, or something beginning to emerge: 日本語が分かってきた (I've gradually come to understand Japanese — up to now), 雨が降ってきた (it's started to rain). The image here is of change "moving toward" the speaker.
Conversely, 〜ていく expresses change from now onward: これから寒くなっていく (it will keep getting colder from now on), これからも勉強していきます (I'll keep studying from here on). Just remember: てくる = toward the present, ていく = toward the future.
These two patterns also keep their spatial meaning with verbs of motion and carrying: 持っていく (take along), 持ってくる (bring), 買ってくる (go buy something and come back) — the action is viewed as moving away from or toward the speaker.
Examples
日本語が少しずつ分かってきました。
I've gradually come to understand Japanese.
これから寒くなっていきますね。
It's going to keep getting colder from now on, isn't it?
急に雨が降ってきました。
It suddenly started to rain.