What duties are Jews required to do?

Answer 1

All the hundreds of mitzvoth (commands), principles and beliefs of the Torah.
See the highlighted Related Links below, for full detail on these.The basic beliefs of Judaism

The practices of Judaism

The principles of Judaism

The ethics of Judaism

How Judaism began

Answer 2

Orthodox Jews believe that Jews must follow halakha (Jewish law).

Non-Orthodox Jews generally regard most of halakha as custom, rather than law.

But in either case, halakha is vitally important to Jews. It's a code of ethics that cover every aspect of Jewish life.

Answer 3

Religious Jews live by the Torah, by obeying its commands and beliefs. There are hundreds of examples.
The laws have various reasons. Some (such as the Passover) serve to reenact or remember events of our history.
Some (such as saying the Shema-prayer) serve to reiterate our belief in God.
Some of the laws (such as those of ritual purity and kosher food) serve to sanctify us.
Some (such as the laws of torts) serve to maintain an orderly and just society.
Some (such as the law against breaking a vow) serve to prevent bad character traits.
Some (such as the command to offer help) serve to engender good character traits.
And all of the commands serve to subjugate us to God's will (especially those commands for which no explanation is easily apparent).

Some examples of the commands:
Putting on Tefillin (a.k.a. phylacteries) in the morning
The sukkah-booth during Sukkot
Avoiding leavened products in Passover
Not eating on Yom Kippur
Not working on the Shabbat
Paying workers on time
Marital rights for one's wife

The Ten Commandments
Counting the days of the Omer
Returning lost objects when feasible
Wearing the tzitzith-garment
Learning Torah
Marrying and having children
Educating one's children in Judaism
Giving tzedakah (charity)
Honoring one's parents
And many more.

Note that the Torah "as is" isn't exactly what Judaism observes. Rather, It's the Torah together with the details provided in the Talmud, which is the Oral Law that was handed down together with the laws of Moses. Otherwise, the verses of the Torah often lack enough detail to be fulfilled as is.

See also:

What are the teachings, practices, principles and beliefs of Judaism?

Answer 4

Following the commandments, charity (tzedaka), improving the world for future generations, repairing the world (tikkun olam)