I would like to know how to express that something is equal in some aspect to
another thing (or person, if that makes a difference), as you would express in
English by saying:

X is as (adjective) as Y

for example: This car is as fast as that car.

It would also be interesting for me to learn how to say:

"is not as (adjective) as" "is nearly as (adj) as" "can/might/will be as (adj)
as" "want to be as (adj) as"
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grammar adjectives comparative-constructions comparison
How can I express that X is as big/small/fast/... as Y?
## Exactly what you're looking for...

One phrase which I believe will help you is 「～と同【おな】じくらい～」. This literally
means "to the same degree as" and seems like it will fit what you're asking
for. Some examples:

>   * 耳【みみ】が顔【かお】と同じくらい大きい。 His ears are as big as his face.
>   * モデルと同じくらいかわいい。 She is as cute as a model.
>

This form will with with i-adjectives, but with na-adjectives, you'll need to
use に, e.g.:

>   * ちっちゃい魚【さかな】もおっきい魚と同じくらいに[上手]{じょうず}に泳【およ】ぐよ。 Small fish swim just as well
> as big fish.
>

## ...And a [better?] alternative

Of course, you might also consider using 「～のように～」. It doesn't capture
_exactly_ the same nuance, but I think it does well enough. It also has the
benefit of sounding much more natural in many scenarios. For example:

>   * 輝【かがや】く瞳【ひとみ】は星【ほし】のようにきれい。 Your shining eyes are beautiful like the
> stars.
>

## Extending the meaning

Both of these phrases are equally versatile when you want to extend their
meanings. You mentioned "not as", "can/might/will be as", and "want to be
as"... all of these are possible by doing what you'd normally do in Japanese
to say these things.

>   * モデルと同じくらいかわいくなりたい。 I want to be as cute as a model. [Note: It's true! I
> do!]
>   * 耳【みみ】が顔【かお】と同じくらい大きくなるだろう。 His eyes will probably be as big as his face.
>   * 人【ひと】は魚と同じくらいに[上手]{じょうず}に泳【およ】げないよ。 A person can't swim as well as a
> fish.
>

And just in case you really want your girlfriend to break up with you:

>   * 瞳【ひとみ】は星【ほし】のようにきれいじゃない。 Your eyes are not beautiful like the stars.
>

Note: basis for example sentences sourced from various Japanese-language
websites.
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answer
In addition to `くらい／ぐらい` as @rintuan mentioned, `ほど` (meaning "extent" or
"limit") is your friend:

> ### X is not as (adjective) as Y → ほど〜ない
>
>   * あの少年はお兄さん **ほど** 背が **高くない** → That boy is not as tall as his older
> brother.
>   * 生物学は物理学 **ほど** 面白く **ない** → Biology is not as interesting as physics.
>

>
> ### Other comparisons: "to the extent of 〜"
>
>   * 今日はセーターを着るほど寒い → Today is cold enough to (need to) wear a sweater;
> "Today is cold to the extent of wearing a sweater"
>

Some other alternatives:

> ### Two things being nearly equal in some aspect
>
>   * この車とあの車と、速さはだいたい同じ → This car is about as fast as that car; "As for this
> car and that car, their speeds are about the same"
>   * ねこくらいの大きさのネズミ → A rat (about) as big as a cat
>
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answer