• Provision of signed written informed consent.

• Solid tumour cohorts only (Phase I and Phase II; excluding NUC-7738 + pembrolizumab cohort): Histologically confirmed diagnosis of an advanced solid tumour with measurable disease as per RECIST v1.1 criteria and/or evaluable disease (evaluable: cytologically or radiologically detectable disease such as ascites, peritoneal deposits, or lesions, which do not fulfil RECIST v1.1 criteria for measurable disease) for solid tumours.

• NUC-7738 + pembrolizumab cohort only (Phase II): Histologically confirmed diagnosis of metastatic cutaneous melanoma with measurable disease as per RECIST v1.1 criteria. Must have progressed on ≤2 prior lines of therapy for advanced/metastatic disease (may include one line in the neoadjuvant/adjuvant setting). At least one prior line must have included a PD-1/PD-L1-containing regimen (either monotherapy or in combination) and the patient must have progressed on this regimen within the last 12 months (may have received another therapy in the interim as long as progression on the PD-1/PD-L1-containing regimen was no more than 12 months ago).

• Patients with documented BRAF V600 mutation status may have received prior BRAF-directed therapy (with or without a MEK inhibitor). These patients should be discussed on a case-by-case basis with the Medical Monitor.

• Lymphoma cohort only (Phase II): Relapsed refractory lymphoma (high grade and low grade B-NHL, Hodgkin's Lymphoma and T-cell lymphomas), which is not amenable to standard of care, is refractory to standard of care or for which no standard of care exists. Patients must have bi-dimensionally measurable disease as per Cheson et al, 2007 criteria for lymphoma.

• For solid tumours in single-agent Phase II cohorts only: patients should have received no more than 3 prior lines of treatment for metastatic disease.

• Age ≥18 years (no upper age limit).

• Eastern Cooperative Oncology Group performance status of 0 or 1.

• Life expectancy of ≥12 weeks.

• Adequate bone marrow, liver, and renal function.

⁃ Ability to comply with protocol requirements.

⁃ Female patients of child-bearing potential must have a negative serum pregnancy test within 3 days prior to the first NUC-7738 administration. All patients of child-bearing potential must agree to practice true abstinence or to use two forms of contraception, one of which must be a highly effective method of contraception, from the time of screening until 6 months after the last dose of study medication.

⁃ Phase I and Phase II dose-confirmation cohorts only: Patient must be willing to undergo a new tumour biopsy at Screening and during therapy on the study. Biopsies are mandatory for patient inclusion, except where taking a biopsy would be associated with unacceptable clinical risk due to the location of the disease. Such patients may be discussed on a case-by-case basis with the study Medical Monitor to determine their eligibility. A prior (archival) biopsy that is less than 3 months old may be substituted for a fresh tumour biopsy at Screening with agreement from the Medical Monitor. From protocol v3.4 onwards, biopsies are no longer required.

⁃ Patients must have been advised to take measures to avoid or minimise exposure of the skin and eyes to UV light, including avoiding sunbathing and visits to the solarium, for the duration of study participation and for a period of 4 weeks following the last dose of study medication.