The official teachings of Judaism approve the death penalty in principle but the standard of proof required for application of death penalty is extremely stringent, and in practice, it has been abolished by various Talmudic decisions, making the situations in which a death sentence could be passed effectively impossible and hypothetical. "Forty years before the destruction" of the Temple in Jerusalem in 70 CE, i.e. in 30 CE, the Sanhedrin effectively abolished capital punishment, making it a hypothetical upper limit on the severity of punishment, fitting in finality for God alone to use, not fallible people.[3]

While allowing for the death penalty in some hypothetical circumstances, scholars of Judaism are broadly opposed to the death penalty as practiced in the modern world. The Jewish understanding of Biblical law is not based on a literal reading of the Bible, bur rather through the lens of Judaism's oral law. These oral laws were first recorded around 200 CE in the Mishnah and later around 600 CE in the Babylonian Talmud. The laws make it clear that the death penalty was used only rarely. The Mishnah states:

A Sanhedrin that puts a man to death once in seven years is called destructive. Rabbi Eliezer ben Azariah says: a Sanhedrin that puts a man to death even once in seventy years. Rabbi Akiba and Rabbi Tarfon say: Had we been in the Sanhedrin none would ever have been put to death (Mishnah, Makkot 1:10).
Rabbinic law developed a detailed system of checks and balances to prevent the execution of an innocent person, and these were so restrictive as to effectively legislate the penalty out of existence. The law requires that:

There must have been two witnesses to the crime, and these must conform to a prescribed list of criteria. For example, females and close relatives of the criminal are precluded from being witnesses according to Biblical law, while full-time gamblers are precluded as a matter of Rabbinical law.
The witnesses must have verbally warned the person that they were liable for the death penalty
The person must then have acknowledged that he or she was warned, and yet then have gone ahead and committed the sin regardless.
No individual was allowed to testify against him or herself.
In law schools everywhere, students read the famous quotation from the 12th Century legal scholar, Maimonides,

"It is better and more satisfactory to acquit a thousand guilty persons than to put a single innocent one to death."
Maimonides argued that executing a defendant on anything less than absolute certainty would lead to a slippery slope of decreasing burdens of proof, until we would be convicting merely "according to the judge's caprice." (Caprice of all kinds are more visible now with computers, statistics, DNA evidence, and new discovery laws directed at prosecutors' files.) Maimonides was concerned about the need for the law to guard itself in public perceptions, to preserve its majesty and retain the people's respect.[4]

Today, the State of Israel only uses the death penalty for extraordinary crimes. The only execution ever to take place in Israel was in 1962, against convicted Nazi war criminal Adolf Eichmann. However, Israeli employment of the death penalty has little to do with Jewish law.

In Orthodox Judaism, it is held that in theory the death penalty is a correct and just punishment for some crimes. However in practice the application of such a punishment can only be carried out by humans whose system of justice is nearly perfect, a situation which has not existed for some time.

Orthodox Rabbi Yosef Edelstein

So, at least theoretically, the Torah can be said to be pro-capital punishment. It is not morally wrong, in absolute terms, to put a murderer to death ...However, things look rather different when we turn our attention to the practical realization of this seemingly harsh legislation. You may be aware that it was exceedingly difficult, in practice, to carry out the death penalty in Jewish society ...I think it's clear that with regard to Jewish jurisprudence, the capital punishment outlined by the Written and Oral Torah, and as carried out by the greatest Sages from among our people (who were paragons of humility and humanity and not just scholarship, needless to say), did not remotely resemble the death penalty in modern America (or Texas). In theory, capital punishment is kosher; it's morally right, in the Torah's eyes. But we have seen that there was great concern—expressed both in the legislation of the Torah, and in the sentiments of some of our great Sages—regarding its practical implementation. It was carried out in ancient Israel, but only with great difficulty. Once in seven years; not 135 in five and a half. (Rabbi Yosef Edelstein, Director of the Savannah Kollel)
Orthodox Rabbi Aryeh Kaplan writes:

In practice, however, these punishments were almost never invoked, and existed mainly as a deterrent and to indicate the seriousness of the sins for which they were prescribed. The rules of evidence and other safeguards that the Torah provides to protect the accused made it all but impossible to actually invoke these penalties…the system of judicial punishments could become brutal and barbaric unless administered in an atmosphere of the highest morality and piety. When these standards declined among the Jewish people, the Sanhedrin...voluntarily abolished this system of penalties (Rabbi Aryeh Kaplan in Handbook of Jewish Thought, Volume II, pp. 170-71).
In Conservative Judaism the death penalty was the subject of a responsum by its Committee on Jewish Law and Standards:

The Talmud ruled out the admissibility of circumstantial evidence in cases which involved a capital crime. Two witnesses were required to testify that they saw the action with their own eyes. A man could not be found guilty of a capital crime through his own confession or through the testimony of immediate members of his family. The rabbis demanded a condition of cool premeditation in the act of crime before their would sanction the death penalty; the specific test on which they insisted was that the criminal be warned prior to the crime, and that the criminal indicate by responding to the warning, that he is fully aware of his deed, but that he is determined to go through with it. In effect this did away with the application of the death penalty. The rabbis were aware of this, and they declared openly that they found capital punishment repugnant to them… There is another reason which argues for the abolition of capital punishment. It is the fact of human fallibility. Too often we learn of people who were convicted of crimes and only later are new facts uncovered by which their innocence is established. The doors of the jail can be opened, in such cases we can partially undo the injustice. But the dead cannot be brought back to life again. We regard all forms of capital punishment as barbaric and obsolete...
Rabbi Ben Zion Bokser, Statement on capital punishment, 1960. Proceedings of the Committee on Jewish Law and Standards 1927-1970, Volume III, p.1537-1538