1、首先是词性的区别：across为介词,而cross为动词.
（动词为“穿过,横穿”,名词为“十字,十字路口”）
2、当然across必须与through 区别开来.across为“横穿”,与“道路”交叉形成“十字”.而through为在立体空间中的“穿过”.如：go through the forest“穿过森林”,go across the street “穿过大街”
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through表示“贯通、直穿、透过、穿过”的意思,即是从一头（边）贯穿到另一头（边）.
例The river runs through our city.这条河流经我们市.
He passed through the hall. 他穿过大厅
across表示“横穿、横过、横渡、横跨”
例：I swam across the Changjiang River 20 years ago.
20年前我横渡了长江.
Look left and right before you go across the street.
过马路时要左右看.
through, across和over的区别
①、through：穿过
I threw it through the window.
Is it quicker to drive round the town, or straight though the centre?
②、across: from one side to the other
（在平面上）越过,在/到……对/面,横过.
I can’t swim across the river.
我不能游到河对岸.
He went across the hall to the door.
他越过大厅到门口.
③、over: 翻越过
across和over两者都表示从一侧向另一侧的运动：
She drove over /across the bridge.
但如果越过的是一个高的物体,则用over:
He climbed over the wall.
如果越过的是一个平面,则宜用across：
He went across the hall to the door.
across和through都表示“穿过”,“通过”的意思,across表示从物体表面“穿过”、“通过”;而through则表示从内部“穿过”、“通过”.
by 的用法
介词,表示通过…方法或途径的意思,译成“靠,通过”,后面可加名词或名词短语.
e.g. 1) The house was destroyed by fire. 房屋被火烧毁了.
2) travel by air /land/sea. 航空（陆路,航海）旅行
3) go by train /boat/ bus 乘火车（船,公共汽车）去
4) shake sb. by the hand 和某人握手
5) I study English by watching English movies. 我通过看英文电影学英语.
另外,by作为介词的意义有很多,我们也已经学过一些用法,总结如下：
1）在…旁边,靠近 e.g. There is a power station. 河边有一个电厂.
2）沿着,经由 e.g. come by the highway 由公路来
3）由于 e.g. by mistake 由于差错
4）被,由 e.g. some articles written by Lu Xun. 一些由鲁迅写的文章
5）表示面积 e.g. a room 5m by 4m 一间长五米宽四米的房间
6）逐批 e.g. one by one 一个接一个
They swam across the river. 他们游过河去.
They walked through the forest. 他们走过森林.