We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for {mathbb {C}}P^2# overline{{mathbb {C}}P^2}, {mathbb {C}}P^2# 2overline{{mathbb {C}}P^2}, we are able to get almost toric base diagrams (ATBDs) of triangular shape and prove the existence of infinitely many symplectomorphism (in particular Hamiltonian isotopy) classes of monotone Lagrangian tori in {mathbb {C}}P^2# koverline{{mathbb {C}}P^2} for k=0,3,4,5,6,7,8. We name these tori Theta ^{n_1,n_2,n_3}_{p,q,r}. Using the work of Karpov-Nogin, we are able to classify all ATBDs of triangular shape. We are able to prove that {mathbb {C}}P^2# overline{{mathbb {C}}P^2} also has infinitely many monotone Lagrangian tori up to symplectomorphism and we conjecture that the same holds for {mathbb {C}}P^2# 2overline{{mathbb {C}}P^2}. Finally, the Lagrangian tori Theta ^{n_1,n_2,n_3}_{p,q,r} subset X can be seen as monotone fibres of ATFs, such that, over its edge lies a fixed anticanonical symplectic torus Sigma . We argue that Theta ^{n_1,n_2,n_3}_{p,q,r} give rise to infinitely many exact Lagrangian tori in X setminus Sigma , even after attaching the positive end of a symplectization to partial (X setminus Sigma ).