This paper seeks to address the question of designing distributed algorithms for the setting of compact memory i.e. sublinear bits working memory for arbitrary connected networks. The nodes in our networks may have much lower internal memory as compared to the number of their possible neighbours implying that a node may not be able to store all the IDs of its neighbours. These algorithms are useful for large networks of small devices such as the Internet of Things, for wireless or ad-hoc networks, and, in general, as memory efficient algorithms. We introduce the Compact Message Passing(CMP) model;an extension of the standard message passing model considered at a finer granularity where a node can interleave reads and writes with internal computations, using a port only once in a round. The interleaving is required for meaningful computations due to the low memory requirement and is akin to a distributed network with nodes executing streaming algorithms. Note that the internal memory size upper bounds the message sizes and hence e.g. for log-memory, the model is weaker than the Congest model; for such models our algorithms will work directly too. We present early results in the CMP model for nodes with log^2-memory. We introduce the concepts of local compact functions and compact protocols and give solutions for some classic distributed problems (leader election, tree constructions and traversals). We build on these to solve the open problem of compact preprocessing for the compact self-healing routing algorithm CompactFTZ posed in Compact Routing Messages in Self-Healing Trees(TCS2017) by designing local compact functions for finding particular subtrees of labeled binary trees. Hence, we introduce the first fully compact self-healing routing algorithm. We also give independent fully compact versions of the Forgiving Tree[PODC08] and Thorup-Zwick's tree based compact routing[SPAA01].