Five-year-olds will learn fractions and computer coding, while those in early secondary school will have to study at least two Shakespeare plays.
The curriculum is being implemented for most year groups simultaneously.
Teachers' leaders say the timetable is unrealistic, but the Department for Education said its aim was to prepare children for "life in modern Britain".
A spokesman said the government wanted "all children to learn the core knowledge in key subjects - the ones universities and employers value the most".
All local authority primary and secondary schools have to start teaching the new national curriculum from the start of term.
It is not compulsory for academies - which are now a majority of secondary schools.
The rewritten national curriculum, described by the prime minister as "rigorous, engaging and tough", sets out the framework for what children should be taught between the ages of five and 14.
Former education secretary Michael Gove has said changes were necessary for England to keep pace with the most successful education systems in the world.
The new-look curriculum puts a stronger emphasis on skills such as "essay writing, problem-solving, mathematical modelling and computer programming".
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