This monograph is an account of the author's investigations of gradient vector flows on compact manifolds with boundary. Many mathematical structures and constructions in the book fit comfortably in the framework of Morse Theory and, more generally, of the Singularity Theory of smooth maps.The geometric and combinatorial structures, arising from the interactions of vector flows with the boundary of the manifold, are surprisingly rich. This geometric setting leads organically to many encounters with Singularity Theory, Combinatorics, Differential Topology, Differential Geometry, Dynamical Systems, and especially with the boundary value problems for ordinary differential equations. This diversity of connections animates the book and is the main motivation behind it.
The book is divided into two parts. The first part describes the flows in three dimensions. It is more pictorial in nature. The second part deals with the multi-dimensional flows, and thus is more analytical. Each of the nine chapters starts with a description of its purpose and main results. This organization provides the reader with independent entrances into different chapters.

Part I: Flows and Spines on 3-manifolds; Vector Fields, Morse Stratifications, and Gradient Spines of 3-folds; Combinatorial and Gradient Complexities of 3-folds; Flow Deformations and Gradient Spines in 3D; Part II: Morse Theory on Manifolds with Boundary; Morse Stratifications and Tangency of Vector Fields to the Boundary; Spaces of Multi-tangent Trajectories; Spines and Flow Spines; Schwartz Genera as a Complexity Measure of Traversing Flows; Spectral Sequences for Spaces of Multi-tangent Trajectories and the Filtration Invariants of Flows; Convexly Enveloped Bordisms of Morse Data and the Complements to Discriminants of Smooth Maps; The Burnside-ring-valued Morse Formula for Vector Fields on Manifolds with Boundary.