The main goal of this lecture is to introduce the notion of a weakly Lipschitz mapping on a fixed Cq stratification, discuss its fundamental properties and give some examples. A natural setting for our results is the theory of o-minimal structures on the ordered field of real numbers R. In definable case we have a theorem about existence of a definable, Lipschitz, weakly bi-Lipschitz triangulation of a relatively compact definable set. We distinguish a class of WL conditions which are in some sense invariant with respect to definable, locally Lipschitz, weakly bi-Lipschitz homeomorphisms. We also define a class of T conditions that involves the WL conditions with a conical property. In particular, the Whitney (B) condition and the Verdier condition belong to the T class. As a final result we have the following triangulation theorem: Let Q be a T condition of class Cq, q ∈ N ∪ {∞, ω}. Let A ⊂ Rn be a relatively compact, definable set and A1, ..., Ar be definable subsets of A. There exists a definable Cq triangulation (K, H) of A, such that the family {H(4) : 4 ∈ K} is a definable Cq stratification with the Q condition of A and is compatible with A1, ..., Ar. Moreover, H : |K| −→ A is a Lipschitz mapping.