The votes are still being counted in Georgia, but a New York Times live forecast analysis based upon votes counted and where the votes remaining to be counted are located is predicting that Democrats will, against all odds, win both U.S. Senate races by thin margins in runoff elections.
This will leave the United States with a Democratic President and Vice President (after what will probably be a tumultuous count of the electoral votes tomorrow in a joint session of Congress whose outcome is nonetheless basically certain), a 50-50 split in the U.S. Senate that gives Democrats control of the Senate by the thinnest possible margin, and a thin majority (reduced from 235 seats immediately after the 2018 election and one special election held after it was held) in the U.S. House of 222 seats where 218 are needed for a majority.
While Biden had a comfortable popular vote lead, he won the marginal state he needed to win the electoral college vote, Wisconsin, by a thin margin of 0.7 percentage points, the margins in the marginal U.S. Senate races will be thin as well (probably about 1.0 percentage points in the race for the marginal seat [UPDATE, the margin was 0.8 percentage points), and the margins in the marginal U.S. House races (the 5th closest) weren't all that large either: ME-2 and NJ-3 were the 5th and 4th closest races and were won by 1.3 percentage points.
By controlling both houses of Congress and the Presidency, Democrats have immense power to change the country for the better on a variety of fronts. But only if they can hold their caucus together unanimously in the Senate and roughly a 98% consensus in the House. It will help to have President Biden to serve as a party leader and focal point for Democrats. The marginal Democrats in the House and Joe Manchin, the marginal Democrat in the Senate, are quite conservative (although there is no longer any overlap between the elected federal legislators of the parties on a liberal-conservative dimension).
Any moves they make will have to overcome a 6-3 conservative supermajority on the U.S. Supreme Court with hard core conservatives holding the swing votes in most cases, and the filibuster in the U.S. Senate for ordinary legislation.
The long term well being of the Democratic Party would and its ideals would be greatly served by admitting the District of Columbia and Puerto Rico (at least) as states before the next election, by abolishing the remaining scraps of the filibuster in the U.S. Senate, by passing voting rights legislation, and by increasing the size of the U.S. Supreme Court to allow a majority of liberal justices to be appointed to it. All of this can be accomplished with bare majorities.
A constructive response to COVID-19 and sensible economic measures (as well as a general return to sanity and normalcy in politics after four years of crazy) could also improve the odds that Democrats will get something done and win over the public as a result in an extremely divided nation.