This very much depends on which goals we assign the New Deal. On one hand, the New Deal very much met its goal if one focuses on the parts where the New Deal provisioned for an expansion and protection of industrial workers. In that context, the New Deal very much did succeed as things like the Wagner Act and the Fair Labor Standards Act of 1938 were passed. On the other hand the New Deal outright failed to address the hardships of agricultural workers and minority workers who were affected by the Depression.

Labor protections offered to industrial workers in steel, coal, and other industries were never extended to agricultural workers and domestic workers, of which a large percentage of were African American. Likewise, the New Deal despite the massive attempts to create labor projects, did not manage to lift the United States out of the Depression. On the societal side, the better society idealized by the New Deal never came into fruition.

Instead the New Deal became an extension of what Ira Katznelson would coin "Affirmative Action for Whites." The New Deal in many ways through labor legislation, administration, and even in the disbursement of aid whether through direct methods or through things like housing loans, essentially protected the socioeconomic status of the white middle class.

By the 1940s, it was clear that the New Deal though mildly helpful in temporarily employing citizens, had done much less than the massive labor requirements necessitated by WWII. For what it's worth, the New Deal wasn't an abject failure. It did attempt to at least provide protections for workers that had long been denied rights to organize and form unions. Though the unions themselves discriminated on both race and occupation, the growth of unions enabled through the Wagner Act and the FLSA in 1938 were crucial in creating an industrialized and organized middle class who would go on to represent the general American interest in the 1950s.

Unfortunately, the New Deal's successes detracted from and strengthened the institutions which would resist against the Civil Rights Movement in the 1960s. The entrenched working middle class who had benefited massively from being a quasi-exclusive organized workforce would prove difficult to persuade even long after the passing of the Civil Rights Act in 1965.

Works Cited:

-Ira Katznelson, When Affirmative Action Was White: An Untold History of Racial Inequality in Twentieth Century America (2005)

-Mark Brilliant, The Color of America Has Changed: How Racial Diversity Shaped Civil Rights Reform in California, 1941-1978 (2010)

The New Deal worked in the very short term, stimulating the United States economy out of depression by increasing the labor force, beginning massive and wide-ranging infrastructure projects, and stabilizing US monetary policy. "The First Hundred Days"—FDR's first 100 days in office—was the period when the majority of these policies and projects were put in place. FDR had immense political capital coming into office, given the unpopularity of Hoover and his policies. FDR cashed this in immediately by rolling out banking and monetary reform as well as the famous public works projects: PWA, CCC, TVA, etc.—all those acronyms.

These initial changes stabilized the US economy and got things back on track. There was NOT an immediate return to the rapid growth enjoyed through the 1920's, but by 1934 things were no longer dire. It was during what is called "The Second New Deal" (1935-38) that FDR was able to get the Social Security Act passed, which of all the New Deal programs had (and still has) the largest affect on the most Americans.

So by the late 1930s, the US and the world was out of the depression. But the New Deal doesn't account for America's longest and most consistent period of growth from 1945-1970. FDR is often credited with laying the foundation for this unprecedented period of economic growth, but it would be more accurate to say that American opportunism in the years immediately following WWII are most responsible for this period. The Marshall Plan (stimulating and sustaining US manufacturing by giving war-ravaged Europe money to buy American goods), dividing Europe with the Soviet Union to guarantee US access to Western European markets, the entrance of American business interests into Europe's former colonies (Africa and the Middle East) and Japan's old colonial stomping grounds (Japan itself, but also later Korea, Taiwan, and SE Asia), and a host of other well-timed policies and economic processes crowned America the economic hegemon of the postwar world.

In short, the New Deal worked to pull the US out of depression and stabilize the national economy in the short term. But FDR is often credited with creating America's huge postwar economic boom, for which he is only partially responsible. This mingling of records—the short term success of the New Deal and the long period of postwar economic growth—has created a historical narrative that charts the US on an upward economic trajectory through the 20th century. Because FDR was in charge and handing down policies when this process was set in motion (early 1930s), he is often credited with all the success that follows.

Sources:

Irresistible Empire, Victoria De Grazia

The Tragedy of American Diplomacy, William Appleman Williams (see chapters 4-6)

/u/buy_a_pork_bun already gave an interesting answer. I don't want to dispute anything he said.

I like to expand on some economic points. When you interpret the question as, "Did the New Deal lift the economy out of depression?", then the answer is definitely no. Since this is what leftists like Sanders like to claim, this is what Im gone assume you are asking about. If it achieves alternative goals is another question and Im not an expert at answering those.

The economic debate on this question is quite old. For a long time it was unquestioned that the New Deal expanded spending did save the economy. In 1963 Milton Friedman and Anna J. Schwartz released a book called "A Monetary History of the United States" and it showed relatively clearly that monetary policy was probably far more important then any of the fiscal factors. This argument (and more modern versions) has slowly won out with most economists. The old interpretation was abandoned even by leftist economists, even if they did not accept the Friedman/Schwartz argument, they shifted their expiation to the spending increase in WW2 instead of the "New Deal".

So if you look at FDRs New Deal, most economist would argue that the most significant thing he did was to abandon the gold standard. That generated 4 month of extremely rapid growth in industrial production, some of the fastest growth in US history. However once NIRA started it dropped of again.

If you compare globally it gives you a lot of context. Most countries were moving out of the depression by 1933 and were out by 1935. The US was extremely exceptional that it still was in depression by 1937. I would argue that what most people call "New Deal" was either neutral or harmful to the recovery.

Here a number of books that agree on the basics, they are from different time periods to illustrate the ongoing scholarship of gold/monetary question:

[1] A Monetary History of the United States

[2] Gold, France, and the Great Depression, 1919-1932

[3] Golden Fetters: The Gold Standard and the Great Depression, 1919-1939

[4] The Midas Paradox: Financial Markets, Government Policy Shocks, and the Great