This clinical trial will assess whether a medicine called artemether, currently used at a lower dose to treat malaria, could also be useful in treating cancer. For this trial, the medicine is given in a new way, as a spray under the tongue. This delivery has several advantages, including better absorption into the body and it may be easier for people to take. It will be prescribed to consenting patients with advanced solid tumours, every day for 4 months, either once or twice a day. (For this study 'advanced tumour' will mean there are no other effective standard therapy options available to the patient).

The dose will be decided in Phase 1 of the study by initially testing 3 different doses of the medicine in up to 21 different cancer patients, to make sure it is not too toxic or causes any side effects.

After this, the highest safe dose identified in the first phase will be tested in up to 66 cancer patients in Phase 2 of the study, to see if the medicine is able to reduce the size of their tumour or slow down its growth.

Inclusion Criteria:

1. Subjects of 18 years or older at the Screening Visit.
2. Histologic and/or cytologically confirmed diagnosis of solid tumour with clinical and imaging evidence that the tumour is advanced and for whom there is no effective standard therapy available.
3. At least one evaluable tumour that is at least 10 mm by computerised tomography (CT) scan or magnetic resonance imaging (MRI) at the pre-treatment stage
4. At least one prior systemic anti-cancer treatment and disease that is refractory or progressive following treatment.
5. Eastern Co operative Oncology Group (ECOG) performance ≤ 2.
6. An adequate renal, liver and bone marrow function.
7. Women of child bearing potential (WOCBP) must have a negative pregnancy test before the start of treatment.
8. Sexually active women of childbearing potential must be using an acceptable form of contraception, as detailed in the protocol.
9. Male subjects must have had a successful vasectomy (confirmed azoospermia) or they and their female partner must meet the criteria defined in the protocol (i.e., not of childbearing potential or practicing highly effective contraception throughout the study period and for 30 days after study drug discontinuation). Those with partners using hormonal contraceptives must also be using an additional approved method of barrier contraception.
10. Expected life expectancy > 4 months.

Exclusion Criteria:

1. Unwillingness or inability to provide informed consent.
2. Any evidence of severe or uncontrolled systemic diseases, infection or laboratory finding that in the view of the Investigator makes it undesirable for the patient to participate in the trial.
3. Allergic to artemether or other artemisinin derivatives or any of the components of the sublingual formulation.
4. Pregnancy or lactation.
5. Subject had major surgery or significant traumatic injury within 4 weeks of start of study drug; subject has not recovered from the side effects of any major surgery (defined as requiring general anaesthesia) or subject might require major surgery during the course of the study.
6. Subject has had anti-cancer treatment within 4 weeks of start of study drug, excluding small field palliative irradiation which may be performed up to 2 weeks prior to start of study drug.
7. Subject has had prior treatment with any investigational drug within the preceding 4 weeks before study start.