The rise is explained almost entirely by the expected increase in the number of people living in the UK and the ageing population.
Cancer Research UK, which funded the study published in the British Journal of Cancer, says the NHS must act now to avoid being "overwhelmed".
The Department of Health said it was already responding to the challenge.
The research presents projections for 23 different types of cancer. The results are based on taking figures from cancer registries going back to 1975, and then projected forwards to 2030, taking into account expected changes in population.
The study suggests that the number of new cases could rise from about 298,000 in 2007 to 432,000 by 2030 - an increase of 45%. The increase in men is forecast to be 55%, and 35% in women.
Drawing on data from the Office for National Statistics, the authors assume a rise in the population from 61m in 2007 to nearly 71m by 2030.
The proportion of elderly people is projected to grow at a faster rate. The figure for over 65s stood at 16% in 2007. By 2030 it is expected to reach 22%.The impact of ageing is starkly reflected in the projected increase in the figures for prostate cancer, which is especially common in older men.
Cases are projected to increase from about 36,000 in 2007 to more than 61,000 by 2030. The authors say even this is probably an under-estimate because in future more men are likely to be tested for the disease.
Although the overall number of cancer cases appears set to rise, the authors conclude that after adjusting for the growing and ageing population, cancer rates are likely to remain "broadly stable". But there are variations within this.
The rate of breast cancer is projected to fall by 7%. The authors attribute this to a recent reduction in the use of hormone replacement therapy, which is a risk factor for the disease.
However the rates of malignant melanoma and kidney cancer are forecast to rise sharply in men and women.
One of the authors, Prof Peter Sasieni acknowledged that the figures should be treated with caution, but said they provided a framework.
"Projections of cancer cases are important for planning health services so we can understand where the future burden is on the NHS and also where health awareness messages need to be raised."
Cancer Research UK chief executive Harpal Kumar added: "At a time when the finances of the health service are being squeezed, it is absolutely crucial that health commissioners plan now for a massive increase in demand for cancer services, to ensure we provide high quality care to all."
The charity's director of health information, Sara Hiom, said: "Smoking and drinking alcohol are two of the biggest things that increase the chance of developing oral, liver and kidney cancer - so by stopping smoking and cutting back on alcohol, we can lower our risk of these cancers as well as other diseases.
"Maintaining a healthy bodyweight is also important in cutting the risk of liver and kidney cancers."
A spokesman for England's Department of Health said the NHS was already responding to the challenge of an ageing population and increases in cancer cases.
"That is why we are investing more than £750m over the next four years to make sure people are diagnosed with cancer earlier and have better access to the latest treatments," he added.