Democratic nomination rivals Hillary Clinton and Bernie Sanders are fighting their latest duel in Wyoming, the US state with the fewest delegates.
The state’s caucuses have opened to elect just 14 delegates, with a further four chosen at a state convention next month.
Mrs Clinton remains far ahead overall despite recent wins for her rival.
In the Republican race, Ted Cruz is tipped to pick up further delegates at the party’s Colorado convention.
In state assemblies which culminated on Friday, Mr Cruz won 21 delegates to just two for Donald Trump, who still has a comfortable lead in the overall nomination race for the Republicans.
On Saturday, a further 13 delegates are at stake at the Colorado Republican Convention.
The next big prize for both parties is the New York primaries on 19 April: 291 delegates are at stake for the Democrats and 95 for Republicans.
Mr Sanders can boast four wins in a row and victories in six of the last seven contests, the latest being Wisconsin.
While a win on Saturday would boost him further, he still has only 1,061 delegates to 1,749 for Mrs Clinton, when the latter’s 469 superdelegates are added to the tally. To win, a candidate needs 2,383.
Mr Cruz has 520 Republican delegates to 743 for Mr Trump. Mr Cruz is hoping to win at least enough votes to block an outright win for Mr Trump and force a decision at the party’s convention in July.
The presidential election itself, on 8 November, will see America vote for a successor to Barack Obama, a Democratic president standing down after two terms in office which have seen the Republicans take control of both houses of Congress.
19 April: New York primary
26 April: Primaries in Pennsylvania, Connecticut, Delaware, Maryland and Rhode Island