Open-Label, Non Randomized Phase 2 Study With Safety Run-In

Inclusion Criteria: 1. Histologically confirmed diagnosis\* of relapsed or refractory lymphoma, received at least two prior lines of therapy including immuno-chemotherapy. Patients with relapsed Chronic Lymphoid Leukemia (CLL) are eligible if they have received one or more prior lines of any approved standard therapy. \* archival biopsies may be used if obtained up to a year prior to enrollment; re-biopsy is strongly recommended if last biopsy was obtained more than a year ago. 2. Only for patients in the Phase 2 part: At least one measurable nodal or extra-nodal lesion defined as follows: Clearly measurable (i.e. well-defined boundaries) in at least two perpendicular dimensions on imaging scan with \> 1.5 cm in longest transverse diameter. 3. Age ≥ 18 years 4. Eastern Cooperative Oncology Group (ECOG) Performance Score of 0-1 (See Appendix 2). 5. Adequate organ system functions defined as: 1. Absolute neutrophil count (ANC) ≥1.0x109/l 2. Platelets ≥ 75x109/l 3. Haemoglobin ≥ 85g/L 4. Adequate hepatic function, defined as Total bilirubin ≤ 1.5 times the upper limit of normal (ULN) and Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) ≤ 2.5 times ULN (or ALT/AST ≤ 5 times ULN in patients with liver involvement) 5. Adequate renal function, defined as serum creatinine ≤ 1.5 times ULN 6. Fasting glucose \< 7.0 mmol/L; Glycated haemoglobin (HbA1c) \< 6.4% 6. Ability and willingness to swallow and retain oral medication. 7. Willingness and ability to comply with the trial procedures 8. Female and male patients with reproductive potential must agree to use effective contraception from screening until 90 days after discontinuation of PQR309 9. Signed informed consent Exclusion Criteria: Any of the following conditions precludes enrollment of a patient: 1. Immunosuppression due to: * Allogeneic hematopoietic stem cell transplant (HSCT) * Any immune-suppressive therapy within 4 weeks prior to trial treatment start * Known HIV infection 2. Autologous stem cell transplant within 3 months prior to trial treatment start. 3. Concomitant anticancer therapy (e.g. chemotherapy, radiotherapy, hormonal therapy, immunotherapy, biological response modifier, signal transduction inhibitors). 4. Concomitant treatment with medicinal products that increase the pH (reduce acidity) of the upper gastrointestinal tract, including, but not limited to, proton-pump inhibitors (e.g. omeprazole), H2-antagonists (e.g. ranitidine) and antacids. Patients may be enrolled in the study after a wash-out period sufficient to terminate their effect. 5. Use of any investigational drug within 21 days prior to trial treatment start. 6. Patients who experienced National Cancer Institute (NCI) Common Terminology Criteria For Adverse Events (CTCAE) ≥ Grade 3 on PI3K/mTOR inhibitors 7. Any major surgery, chemotherapy or immunotherapy within 21 days prior to trial treatment start. 8. Symptomatic or progressing Central nervous system (CNS) involvement. Exception: Patients with meningeal involvement can be included upon discussion between the sponsor and the investigator. 9. Persisting toxicities NCI CTCAE ≥2 related to prior anticancer therapy 10. Presence of gastrointestinal disease or any other condition that could interfere significantly with the absorption of the study drug. 11. Severe/unstable angina, myocardial infarction or coronary artery bypass within the last 3 years prior to trial treatment start, symptomatic congestive heart failure New York Heart Association (NYHA) Class 3 or 4, hypertension BP\>150/100mmHg 12. A serious active infection at the time of treatment, or another serious underlying medical condition that could impair the ability of the patient to receive treatment. 13. Lack of appropriate contraceptive measures (male and female) 14. Pregnant or lactating women 15. Known HIV infection 16. Significant medical conditions which could jeopardize compliance with the protocol. 17. Uncontrolled diabetes mellitus; patients with controlled diabetes may be enrolled (see fasting glucose and HbA1c levels in inclusion criteria).